Derivatives

Problem 1

Differentiate $x^{e^{x}}$ with respect to $x$ .

See Solution

Problem 2

Find the derivative of $x^{2}$ with respect to $x$ .

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Problem 3

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t$ .

See Solution

Problem 4

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

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Problem 5

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

See Solution

Problem 6

Find $f(1)$ given $f^{\prime}(x)=3 x^{2}$ and $f(2)=3$ . Choices: (a) -7 (b) -4 (c) 7 (d) 10

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Problem 7

17) Find the area of the region between $y=2x$ and $y=x^2$ . 18) If a cube's side length $S=2$ in decreases at 5 in³/min, find the rate of change of $S$ in in/min.

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Problem 8

Find the derivative $\frac{d}{d x}(\tan (\ln (x)))$ . What is $\frac{d y}{d x}$ for $\sqrt{x}+y^{2}=x y+2$ at (4,0)?

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Problem 9

A particle's velocity is $v(t)=e^{t-1}-3 \sin (t-1)$ . What is its motion at $t=1$ ? (a) speeding up, (b) slowing down, (c) neither, (d) at rest.

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Problem 10

At $t=2$ , what does $R^{\prime}(2)=4$ mean? (a) Depth is 4 inches. (b) Rate is 4 in/hr. (c) Rate of change is 4 in/hr². (d) Depth increased by 4 inches.

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Problem 11

Find the $x$ -coordinate of the inflection point of $f(x)=\int_{1}^{x}(3t-6t^{2})dt$ . Options: (a) $-\frac{1}{4}$ (b) $\frac{1}{4}$ (c) 0 (d) $\frac{1}{2}$ . Evaluate $\int \sin(3x)dx$ . Options: (a) $3\cos(3x)+C$ (b) $\frac{1}{3}\cos(3x)+C$ (c) $-3\cos(3x)+C$ (d) $-\frac{1}{3}\cos(3x)+C$ . Count removable discontinuities of $y=\frac{x+2}{x^{4}+16}$ . Options: (a) one (b) two (c) three (d) four.

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Problem 12

Find $f'(x)$ if $f(x)=\frac{x^{2}+1}{3 x}$ and evaluate $\int(2 x+3)(x^{2}+3 x)^{4} dx$ .

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Problem 13

Find the ball's instantaneous velocity at $t=10.0$ s given its position $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ .

See Solution

Problem 14

Find the derivative of the function $100+(t-3)^{4}(t-5)^{2}$ with respect to $t$ .

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Problem 15

Find the derivative of $y$ with respect to $x$ for the expression $x+\ln |x-3|+\frac{2}{x-3}$ .

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Problem 16

Find the formula for the marginal average profit of the profit function $P(x)=100 \ln (2 x+1)-5 x-10$ . Options are provided.

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Problem 17

Find the derivatives of these functions: 1. $f(x)=6 x^{3}-9 x+4$ , 2. $y=2 t^{4}- t^{2}+13 t$ , 3. $g(z)=4 z^{7}-3 z^{-7}+9 z$ .

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Problem 18

Find $\frac{d}{d x} f^{-1}(0)$ for $f(x)=x+\sin x$ . At $x=0$ , $f$ has slope 2, so $f^{-1}$ slope is $\frac{1}{2}$ .

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Problem 19

Find the derivative of $f(x) = x^2$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 20

Find $\frac{d y}{d x}$ for the equation $2 \cdot x \cdot \ln (y) + y^{2} = 5$ .

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Problem 21

Find $\frac{d y}{d x}$ if $e^{x \cdot y}+5 \cdot y^{4}=2 \cdot y$ .

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Problem 22

Find the marginal profit $P^{\prime}(x)$ for $P(x)=3 \cdot x-8 \cdot \sqrt{x}$ and calculate $\frac{P^{\prime}(x)}{x}$ .

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Problem 23

Find the marginal cost for businesses A and B with $C_A(x)=350+40x+0.12x^2$ and $C_B(x)=280+30x+0.09x^2$ . For $x=500$ , which is cheaper?

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Problem 24

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ .

See Solution

Problem 25

A company sells $Q(x)=60x-x^{2}$ units after spending $\$x$ thousand on ads. Find $Q(7)$ and $Q^{\prime}(7)$ .

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Problem 26

Find the change in demand $x$ when price $p$ drops from \ $2.25 to \$1 using$ x=\frac{1000}{\sqrt{p}+1} $. Also, calculate$ \frac{\Delta X}{\Delta P}$.

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Problem 27

Find the derivative of $(x+1) \sin x$ . What is it?

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Problem 28

Find the derivative of $f(x)=4 \cos 2 x+\log x+x$ . Options: $8 \sin 2 x+1 / x+x$ , $-8 \sin 2 x+1 / x+x$ , $-8 \sin 2 x+1 / x+1$ , $8 \sin 2 x+1 / x+1$ .

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Problem 29

Find the rate of price change when $x=100$ and supply decreases at 8 crates/day from $p x - 20 p - 6 x + 40 = 0$ .

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Problem 30

Differentiate the function: $y = \sqrt{\frac{\sqrt{x}+\frac{1}{\sqrt{x}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}}$

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Problem 31

Find the derivative of $2^{x+2} + 3^{y+1} = 18$ with respect to $x$ .

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Problem 32

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

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Problem 33

Find the $x$ values where the function $f$ has a relative maximum given $f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}$ .

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Problem 34

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

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Problem 35

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

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Problem 36

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

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Problem 37

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

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Problem 38

Estimate $y$ at $x=3.1$ using the tangent line of the curve $x^{3}+x \tan y=27$ at $(3,0)$ . Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

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Problem 39

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

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Problem 40

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

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Problem 41

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

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Problem 42

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

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Problem 43

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

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Problem 44

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

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Problem 45

Find $y^{\prime}(4)$ using implicit differentiation for $4 \cdot x^{2}+3 \cdot x+x \cdot y=4$ and $y(4)=-18$ .

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Problem 46

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

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Problem 47

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

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Problem 48

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

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Problem 49

Find the slope of the tangent line to the curve $4xy^{3}+3xy=7$ at the point $(1,1)$ using implicit differentiation.

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Problem 50

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

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Problem 51

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

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Problem 52

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

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Problem 53

Find the value of $x$ that satisfies the equation $\frac{d}{d x}(x \cdot(967-25-x))=0$ .

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Problem 54

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

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Problem 55

Helium fills a balloon at 2 ft³/s. Find the radius increase rate (in ft/s) after 7 minutes. Use $V=\frac{4}{3} \cdot \pi \cdot r^{3}$ .

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Problem 56

Find the average rate of change of profit $P(x)=-400x^2+6800x-12000$ over $[6,6+h]$ for $h=1, 0.1, 0.01, 0.001, 0.0001$ . What do the results indicate?

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Problem 57

A particle's position is given by $s=2 \cdot t^{3}-7 \cdot t^{2}-9 \cdot t+12$ . Find velocity, acceleration, and specific values.

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Problem 58

Projectile height is given by $h(t)=-16 \cdot t^{2}+256 \cdot t$ . Find average velocity, speed, max height, and acceleration at $t=5$ .

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Problem 59

Differentiate $y=\frac{1}{\sqrt{a+x^{2}}}$ with respect to $x$ .

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Problem 60

Differentiate $y=\sqrt{a+x}$ .

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Problem 61

Find the derivative of $y=x^{3} \ln (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

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Problem 62

Find the tangent line to $f(x)=-x^{2}+5$ at $x=5$ and use it to estimate $f(5.1)$ .

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Problem 63

Find points $(x, y)$ where $\nabla f(x, y)=(0,0)$ for $f(x, y)=\frac{x^{3}}{3}-\frac{y^{3}}{3}+3xy$ and classify them.

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Problem 64

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $\sqrt{y}+x y=5$ .

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Problem 65

Find $\frac{d y}{d x}$ in terms of $x$ given that $x=\sec 3 y$ .

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Problem 66

Find $\frac{d y}{d x}$ in terms of $t$ for $x=\sin(t)$ and $y=\cos^2(t)$ . Simplify your expression.

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Problem 67

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin^3 t$ and $y=\cos^4 t$ .

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Problem 68

Find and simplify $\frac{d y}{d x}$ using the parametric equations $x=\sin(t)$ and $y=\cos^2(t)$ .

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Problem 69

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin(t)$ , $y=\cos^2(t)$ .

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Problem 70

Find the rate of change of sales $S(t) = 10,000 + 2000t - 200t^2$ at $t=0$ , $t=4$ , and $t=8$ weeks.

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Problem 71

Monthly sales (in thousands) for a music store are given by $S(t)=\frac{200}{t^{2}+36}$ . Find $S(2)$ , $S'(2)$ , and estimate sales for month 3.

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Problem 72

Monthly sales for a record album are given by $S(t)=\frac{200 t}{t^{2}+36}$ . Find: a) $S(2)$ , b) $S^{\prime}(2)$ , c) estimate sales in month 3 using (a) and (b).

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Problem 73

Differentiate $x^{3} y^{2}$ with respect to $y$ .

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Problem 74

Differentiate $x^{3} y^{2}$ with respect to $x$ .

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Problem 75

Find the free extrema of the function $z=-2x^{2}+22y^{2}-192y+x^{2}y$ .

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Problem 76

Find the free extremes of the function $z=3 x^{2}-8 y^{2}+64 y-y x^{2}$ .

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Problem 77

Find $\frac{d y}{d x}$ for the equation $4 e^{-3 x}+e^{4 y}+8=0$ .

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Problem 78

Differentiate the function $\frac{5 x^{2}-5}{2 x}$ .

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Problem 79

Differentiate the function $F(x)=\frac{\ln x}{3 x^{2}}-\frac{e^{-9 x}}{\ln x^{2}}$ .

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Problem 80

Find the second derivative $H^{\prime \prime}(x)$ of the function $H(x)=x e^{x}$ .

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Problem 81

Find the derivative $g^{\prime}(x)$ for the function $g(x)=\frac{x^{2}}{1+x}+\frac{e^{x}}{1+x^{2}}$ .

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Problem 82

An ice sphere melts at $2 \pi$ m³/hr. Find the surface area decrease rate when radius is 5 m. Options: (A) $\frac{4 \pi}{5}$ (B) $40 \pi$ (C) $80 \pi^{2}$ (D) $100 \pi$

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Problem 83

Find the derivative of the function $\frac{e^{x}}{2^{e^{x}}}$ with respect to $x$ .

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Problem 84

Find $f(x)$ if $\frac{d f}{d x}=3 x^{2}-3-4 e^{2 x}$ and $f(0)=8$ . Choose from options a, b, c, d.

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Problem 85

What is the derivative of $\cos x$ with respect to $x$ ?

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Problem 86

A company sells $Q(x) = 60x - x^2$ units after spending $\$x$ thousand on ads. Find $Q(7)$ and $Q'(7)$ , and interpret.

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Problem 87

Simplify the expression $\left(\frac{04 y^{8}}{y^{\prime}}\right)^{\prime}$ .

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Problem 88

Find $f(3)$ given that $f^{\prime}(x) = \frac{x-8}{x-4}$ for $x \neq 4$ .

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Problem 89

Find the derivative of the function $f(x)=2 x^{-10}+7 x^{-2}$ . What is $f^{\prime}(x)$ ?

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Problem 90

Find the partial derivative of $f(x, y)=x^3 y^2+3x e^y$ with respect to $y$ . Options: a, b, c.

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Problem 91

Find the first order partial derivative of $f(x, y)=x^3 y^2 + 3x e^y$ with respect to $y$ .

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Problem 92

Find the derivative of $y=(2 x+9)(x^{3}-1)$ , i.e., calculate $\frac{d y}{d x}$ .

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Problem 93

Calculate the derivative $\frac{d y}{d x}$ for the function $y=\frac{x}{1-5 x}$ .

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Problem 94

Find the derivative $\frac{d y}{d x}$ for $y=(3 x^{2}+5)^{10}$ .

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Problem 95

Find the derivative of $y=\frac{3}{(2 x-1)^{4}}$ with respect to $x$ : $\frac{d y}{d x}$ .

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Problem 96

Differentiate the function $y=(3x+1)^{5}$ .

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Problem 97

Differentiate $y=\left(3 x^{3}-\frac{2}{x}\right)^{2}$ with respect to $x$ .

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Problem 98

Find the tangent line equation for $f(x)=\sqrt{3-x}$ at a given point.

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Problem 99

Find the derivative of the function $y=x^{4}+12x$ . What is $\frac{dy}{dx}$ ?

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Problem 100

A ball is thrown with a velocity of $41 \mathrm{ft} / \mathrm{s}$ . Its height is $y=41 t-22 t^{2}$ . Find average velocity from $t=2$ and instantaneous velocity at $t=2$ .

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Derivatives

Problem 1

Differentiate xex x^{e^{x}} xex with respect to x x x.

Problem 2

Find the derivative of x2 x^{2} x2 with respect to x x x.

Problem 3

Find the ball's instantaneous velocity at t=10.0 t=10.0 t=10.0 s given x(t)=0.000015t5−0.004t3+0.4t x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t x(t)=0.000015t5−0.004t3+0.4t.

Problem 4

A ball's position is given by x(t)=0.000015t5−0.004t3+0.4tx(t)=0.000015 t^5 - 0.004 t^3 + 0.4 tx(t)=0.000015t5−0.004t3+0.4t. Find its velocity at t=10.0t=10.0t=10.0 s.

Problem 5

A ball's position is given by x(t)=0.000015t5−0.004t3+0.4tx(t)=0.000015 t^5 - 0.004 t^3 + 0.4 tx(t)=0.000015t5−0.004t3+0.4t. Find its velocity at t=10.0t=10.0t=10.0 s.

Problem 6

Find f(1) f(1) f(1) given f′(x)=3x2 f^{\prime}(x)=3 x^{2} f′(x)=3x2 and f(2)=3 f(2)=3 f(2)=3. Choices: (a) -7 (b) -4 (c) 7 (d) 10

Problem 7

17) Find the area of the region between y=2x y=2x y=2x and y=x2 y=x^2 y=x2. 18) If a cube's side length S=2 S=2 S=2 in decreases at 5 in³/min, find the rate of change of S S S in in/min.

Problem 8

Find the derivative ddx(tan⁡(ln⁡(x))) \frac{d}{d x}(\tan (\ln (x))) dxd​(tan(ln(x))). What is dydx \frac{d y}{d x} dxdy​ for x+y2=xy+2 \sqrt{x}+y^{2}=x y+2 x​+y2=xy+2 at (4,0)?

Problem 9

A particle's velocity is v(t)=et−1−3sin⁡(t−1) v(t)=e^{t-1}-3 \sin (t-1) v(t)=et−1−3sin(t−1). What is its motion at t=1 t=1 t=1? (a) speeding up, (b) slowing down, (c) neither, (d) at rest.

Problem 10

At t=2 t=2 t=2, what does R′(2)=4 R^{\prime}(2)=4 R′(2)=4 mean? (a) Depth is 4 inches. (b) Rate is 4 in/hr. (c) Rate of change is 4 in/hr². (d) Depth increased by 4 inches.

Problem 11

Problem 12

Find f′(x) f'(x) f′(x) if f(x)=x2+13x f(x)=\frac{x^{2}+1}{3 x} f(x)=3xx2+1​ and evaluate ∫(2x+3)(x2+3x)4dx \int(2 x+3)(x^{2}+3 x)^{4} dx ∫(2x+3)(x2+3x)4dx.

Problem 13

Find the ball's instantaneous velocity at t=10.0t=10.0t=10.0 s given its position x(t)=0.000015t5−0.004t3+0.4tx(t)=0.000015 t^5 - 0.004 t^3 + 0.4 tx(t)=0.000015t5−0.004t3+0.4t.

Problem 14

Find the derivative of the function 100+(t−3)4(t−5)2100+(t-3)^{4}(t-5)^{2}100+(t−3)4(t−5)2 with respect to ttt.

Problem 15

Find the derivative of yyy with respect to xxx for the expression x+ln⁡∣x−3∣+2x−3x+\ln |x-3|+\frac{2}{x-3}x+ln∣x−3∣+x−32​.

Problem 16

Find the formula for the marginal average profit of the profit function P(x)=100ln⁡(2x+1)−5x−10 P(x)=100 \ln (2 x+1)-5 x-10 P(x)=100ln(2x+1)−5x−10. Options are provided.

Problem 17

Find the derivatives of these functions: 1. f(x)=6x3−9x+4f(x)=6 x^{3}-9 x+4f(x)=6x3−9x+4, 2. y=2t4−t2+13ty=2 t^{4}- t^{2}+13 ty=2t4−t2+13t, 3. g(z)=4z7−3z−7+9zg(z)=4 z^{7}-3 z^{-7}+9 zg(z)=4z7−3z−7+9z.

Problem 18

Find ddxf−1(0)\frac{d}{d x} f^{-1}(0)dxd​f−1(0) for f(x)=x+sin⁡xf(x)=x+\sin xf(x)=x+sinx. At x=0x=0x=0, fff has slope 2, so f−1f^{-1}f−1 slope is 12\frac{1}{2}21​.

Problem 19

Find the derivative of f(x)=x2f(x) = x^2f(x)=x2 using the limit definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​.

Problem 20

Find dydx\frac{d y}{d x}dxdy​ for the equation 2⋅x⋅ln⁡(y)+y2=52 \cdot x \cdot \ln (y) + y^{2} = 52⋅x⋅ln(y)+y2=5.

Problem 21

Find dydx\frac{d y}{d x}dxdy​ if ex⋅y+5⋅y4=2⋅ye^{x \cdot y}+5 \cdot y^{4}=2 \cdot yex⋅y+5⋅y4=2⋅y.

Problem 22

Find the marginal profit P′(x)P^{\prime}(x)P′(x) for P(x)=3⋅x−8⋅xP(x)=3 \cdot x-8 \cdot \sqrt{x}P(x)=3⋅x−8⋅x​ and calculate P′(x)x\frac{P^{\prime}(x)}{x}xP′(x)​.

Problem 23

Find the marginal cost for businesses A and B with CA(x)=350+40x+0.12x2C_A(x)=350+40x+0.12x^2CA​(x)=350+40x+0.12x2 and CB(x)=280+30x+0.09x2C_B(x)=280+30x+0.09x^2CB​(x)=280+30x+0.09x2. For x=500x=500x=500, which is cheaper?

Problem 24

Find the ball's instantaneous velocity at t=10.0t=10.0t=10.0 s given x(t)=0.000015t5−0.004t3+0.4tx(t)=0.000015 t^5 - 0.004 t^3 + 0.4 tx(t)=0.000015t5−0.004t3+0.4t.

Problem 25

A company sells Q(x)=60x−x2Q(x)=60x-x^{2}Q(x)=60x−x2 units after spending $x\$x$x thousand on ads. Find Q(7)Q(7)Q(7) and Q′(7)Q^{\prime}(7)Q′(7).

Problem 26

Find the change in demand xxx when price ppp drops from \2.25to$1using2.25 to \$1 using 2.25to$1usingx=\frac{1000}{\sqrt{p}+1}.Also,calculate. Also, calculate .Also,calculate\frac{\Delta X}{\Delta P}$.

Problem 27

Find the derivative of (x+1)sin⁡x(x+1) \sin x(x+1)sinx. What is it?

Problem 28

Problem 29

Find the rate of price change when x=100x=100x=100 and supply decreases at 8 crates/day from px−20p−6x+40=0p x - 20 p - 6 x + 40 = 0px−20p−6x+40=0.

Problem 30

Differentiate the function: y=x+1xx−1xy = \sqrt{\frac{\sqrt{x}+\frac{1}{\sqrt{x}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}}y=x​−x​1​x​+x​1​​​

Problem 31

Find the derivative of 2x+2+3y+1=182^{x+2} + 3^{y+1} = 182x+2+3y+1=18 with respect to xxx.

Problem 32

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=∣x∣f(x)=|x|f(x)=∣x∣ for x>0x>0x>0, x<0x<0x<0, and at x=0x=0x=0. Explain why it's undefined at 000.

Problem 33

Find the xxx values where the function fff has a relative maximum given f′(x)=x2(x+1)3(x−4)2f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}f′(x)=x2(x+1)3(x−4)2.

Problem 34

Find the value of ccc such that f(x)=x+cxf(x)=x+\frac{c}{x}f(x)=x+xc​ has a local minimum at x=3x=3x=3.

Problem 35

Given values of f′′(x)f^{\prime \prime}(x)f′′(x): at x=−1x=-1x=−1 is -4, x=0x=0x=0 is -1, x=1x=1x=1 is 2, x=2x=2x=2 is 5, x=3x=3x=3 is 8. What type of function is fff? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

Problem 36

Find the time ttt (from 0 to 10) when the object with velocity v(t)=tcos⁡t−ln⁡(t+2)v(t)=t \cos t-\ln (t+2)v(t)=tcost−ln(t+2) reaches maximum speed.

Problem 37

Find the time ttt in [0,10][0, 10][0,10] when the velocity v(t)=tcos⁡t−ln⁡(t+2)v(t)=t \cos t - \ln(t+2)v(t)=tcost−ln(t+2) is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

Problem 38

Estimate yyy at x=3.1x=3.1x=3.1 using the tangent line of the curve x3+xtan⁡y=27x^{3}+x \tan y=27x3+xtany=27 at (3,0)(3,0)(3,0). Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

Problem 39

Find the rate of change of sales S(t)=10,000+2000t−200t2S(t)=10,000+2000 t-200 t^{2}S(t)=10,000+2000t−200t2 and interpret S′(t)S^{\prime}(t)S′(t).

Problem 40

Find the derivative of the function f(x)=6⋅x⋅(x−5)f(x)=6 \cdot \sqrt{x} \cdot(x-5)f(x)=6⋅x​⋅(x−5). What is f′(x)f^{\prime}(x)f′(x)?

Problem 41

Differentiate $x^{e^{x}}$ with respect to $x$ .

Find the derivative of $x^{2}$ with respect to $x$ .

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^{5}-0.004 t^{3}+0.4 t$ .

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

A ball's position is given by $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ . Find its velocity at $t=10.0$ s.

Find $f(1)$ given $f^{\prime}(x)=3 x^{2}$ and $f(2)=3$ . Choices: (a) -7 (b) -4 (c) 7 (d) 10

17) Find the area of the region between $y=2x$ and $y=x^2$ . 18) If a cube's side length $S=2$ in decreases at 5 in³/min, find the rate of change of $S$ in in/min.

Find the derivative $\frac{d}{d x}(\tan (\ln (x)))$ . What is $\frac{d y}{d x}$ for $\sqrt{x}+y^{2}=x y+2$ at (4,0)?

A particle's velocity is $v(t)=e^{t-1}-3 \sin (t-1)$ . What is its motion at $t=1$ ? (a) speeding up, (b) slowing down, (c) neither, (d) at rest.

At $t=2$ , what does $R^{\prime}(2)=4$ mean? (a) Depth is 4 inches. (b) Rate is 4 in/hr. (c) Rate of change is 4 in/hr². (d) Depth increased by 4 inches.

Find $f'(x)$ if $f(x)=\frac{x^{2}+1}{3 x}$ and evaluate $\int(2 x+3)(x^{2}+3 x)^{4} dx$ .

Find the ball's instantaneous velocity at $t=10.0$ s given its position $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ .

Find the derivative of the function $100+(t-3)^{4}(t-5)^{2}$ with respect to $t$ .

Find the derivative of $y$ with respect to $x$ for the expression $x+\ln |x-3|+\frac{2}{x-3}$ .

Find the formula for the marginal average profit of the profit function $P(x)=100 \ln (2 x+1)-5 x-10$ . Options are provided.

Find the derivatives of these functions: 1. $f(x)=6 x^{3}-9 x+4$ , 2. $y=2 t^{4}- t^{2}+13 t$ , 3. $g(z)=4 z^{7}-3 z^{-7}+9 z$ .

Find $\frac{d}{d x} f^{-1}(0)$ for $f(x)=x+\sin x$ . At $x=0$ , $f$ has slope 2, so $f^{-1}$ slope is $\frac{1}{2}$ .

Find the derivative of $f(x) = x^2$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

Find $\frac{d y}{d x}$ for the equation $2 \cdot x \cdot \ln (y) + y^{2} = 5$ .

Find $\frac{d y}{d x}$ if $e^{x \cdot y}+5 \cdot y^{4}=2 \cdot y$ .

Find the marginal profit $P^{\prime}(x)$ for $P(x)=3 \cdot x-8 \cdot \sqrt{x}$ and calculate $\frac{P^{\prime}(x)}{x}$ .

Find the marginal cost for businesses A and B with $C_A(x)=350+40x+0.12x^2$ and $C_B(x)=280+30x+0.09x^2$ . For $x=500$ , which is cheaper?

Find the ball's instantaneous velocity at $t=10.0$ s given $x(t)=0.000015 t^5 - 0.004 t^3 + 0.4 t$ .

A company sells $Q(x)=60x-x^{2}$ units after spending $\$x$ thousand on ads. Find $Q(7)$ and $Q^{\prime}(7)$ .

Find the change in demand $x$ when price $p$ drops from \ $2.25 to \$1 using$ x=\frac{1000}{\sqrt{p}+1} $. Also, calculate$ \frac{\Delta X}{\Delta P}$.

Find the derivative of $(x+1) \sin x$ . What is it?

Find the rate of price change when $x=100$ and supply decreases at 8 crates/day from $p x - 20 p - 6 x + 40 = 0$ .

Differentiate the function: $y = \sqrt{\frac{\sqrt{x}+\frac{1}{\sqrt{x}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}}$

Find the derivative of $2^{x+2} + 3^{y+1} = 18$ with respect to $x$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=|x|$ for $x>0$ , $x<0$ , and at $x=0$ . Explain why it's undefined at $0$ .

Find the $x$ values where the function $f$ has a relative maximum given $f^{\prime}(x)=x^{2}(x+1)^{3}(x-4)^{2}$ .

Find the value of $c$ such that $f(x)=x+\frac{c}{x}$ has a local minimum at $x=3$ .

Given values of $f^{\prime \prime}(x)$ : at $x=-1$ is -4, $x=0$ is -1, $x=1$ is 2, $x=2$ is 5, $x=3$ is 8. What type of function is $f$ ? (A) linear (B) quadratic (C) cubic (D) fourth-degree (E) exponential

Find the time $t$ (from 0 to 10) when the object with velocity $v(t)=t \cos t-\ln (t+2)$ reaches maximum speed.

Find the time $t$ in $[0, 10]$ when the velocity $v(t)=t \cos t - \ln(t+2)$ is maximized. Options: A. 9.5 B. 5.1 C. 6.4 D. 7.6

Estimate $y$ at $x=3.1$ using the tangent line of the curve $x^{3}+x \tan y=27$ at $(3,0)$ . Choose from (A) -2.7 (B) -0.9 (C) 0 (D) 0.1 (E) 3.0.

Find the rate of change of sales $S(t)=10,000+2000 t-200 t^{2}$ and interpret $S^{\prime}(t)$ .

Find the derivative of the function $f(x)=6 \cdot \sqrt{x} \cdot(x-5)$ . What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ of the function $f(x)=-3 \cdot \ln (4 \cdot x)$ and calculate $f^{\prime}(2)$ .

Find the derivative of $f(x)=\frac{\sqrt{x}-7}{\sqrt{x}+7}$ and calculate $f^{\prime}(5)$ .

Find the derivative $f'(x)$ of the function $f(x)=\sqrt{4x^2 + 4x + 4}$ and evaluate it at $x=4$ .

Calculate the derivative of $g(x)=(4x^{2}+5x) \cdot e^{x}$ .

Find $y^{\prime}(4)$ using implicit differentiation for $4 \cdot x^{2}+3 \cdot x+x \cdot y=4$ and $y(4)=-18$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=2+\frac{7}{x}+\frac{7}{x^{2}}$ .

Find the derivative of $R(j)=\left(\ln \left(j^{2}\right)\right)^{2}$ at $j=e$ .

Find the derivative $f^{\prime}(t)$ of the function $f(t)=(t^{2}+5t+3)(2t^{-2}+6t^{-5})$ .

Find the slope of the tangent line to the curve $4xy^{3}+3xy=7$ at the point $(1,1)$ using implicit differentiation.

Find the marginal cost for both businesses given their cost functions: $C_A(x)=200+25x+0.1x^2$ and $C_B(x)=400+80x+0.06x^2$ . For $x=500$ , which business has the lowest cost for the next tire?

Find the critical number of the function $f(x)=(8 \cdot x-7) \cdot e^{5 \cdot x}$ .

The demand function is $D(x)=967-25-x$ .
(a) Find the elasticity of demand. (b) Find the price where elasticity equals 1. (c) Determine the price for maximum revenue.

Find the value of $x$ that satisfies the equation $\frac{d}{d x}(x \cdot(967-25-x))=0$ .

Projectile height is given by $h(t)=-16 \cdot t^{2}+256-t$ . Find: (a) average velocity for 3s, (b) speed & height at 6s, (c) max height & time, (d) acceleration at $t=5$ .

Helium fills a balloon at 2 ft³/s. Find the radius increase rate (in ft/s) after 7 minutes. Use $V=\frac{4}{3} \cdot \pi \cdot r^{3}$ .

Find the average rate of change of profit $P(x)=-400x^2+6800x-12000$ over $[6,6+h]$ for $h=1, 0.1, 0.01, 0.001, 0.0001$ . What do the results indicate?

A particle's position is given by $s=2 \cdot t^{3}-7 \cdot t^{2}-9 \cdot t+12$ . Find velocity, acceleration, and specific values.

Projectile height is given by $h(t)=-16 \cdot t^{2}+256 \cdot t$ . Find average velocity, speed, max height, and acceleration at $t=5$ .

Differentiate $y=\frac{1}{\sqrt{a+x^{2}}}$ with respect to $x$ .

Differentiate $y=\sqrt{a+x}$ .

Find the derivative of $y=x^{3} \ln (x)$ with respect to $x$ : $\frac{d y}{d x}$ .

Find the tangent line to $f(x)=-x^{2}+5$ at $x=5$ and use it to estimate $f(5.1)$ .

Find points $(x, y)$ where $\nabla f(x, y)=(0,0)$ for $f(x, y)=\frac{x^{3}}{3}-\frac{y^{3}}{3}+3xy$ and classify them.

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $\sqrt{y}+x y=5$ .

Find $\frac{d y}{d x}$ in terms of $x$ given that $x=\sec 3 y$ .

Find $\frac{d y}{d x}$ in terms of $t$ for $x=\sin(t)$ and $y=\cos^2(t)$ . Simplify your expression.

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin^3 t$ and $y=\cos^4 t$ .

Find and simplify $\frac{d y}{d x}$ using the parametric equations $x=\sin(t)$ and $y=\cos^2(t)$ .

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin(t)$ , $y=\cos^2(t)$ .

Find the rate of change of sales $S(t) = 10,000 + 2000t - 200t^2$ at $t=0$ , $t=4$ , and $t=8$ weeks.

Monthly sales (in thousands) for a music store are given by $S(t)=\frac{200}{t^{2}+36}$ . Find $S(2)$ , $S'(2)$ , and estimate sales for month 3.

Monthly sales for a record album are given by $S(t)=\frac{200 t}{t^{2}+36}$ . Find: a) $S(2)$ , b) $S^{\prime}(2)$ , c) estimate sales in month 3 using (a) and (b).

Differentiate $x^{3} y^{2}$ with respect to $y$ .

Differentiate $x^{3} y^{2}$ with respect to $x$ .