Solved on Feb 06, 2024

Find the derivative of $y=4x^2+2\sqrt{x}$ evaluated at $x=1$ .

STEP 1

Assumptions
1. The given function is $y = 4x^2 + 2\sqrt{x}$ .
2. We need to find the derivative of $y$ with respect to $x$ , denoted as $\frac{dy}{dx}$ .
3. After finding the derivative, we will evaluate $\frac{dy}{dx}$ at $x = 1$ .

STEP 2

To find the derivative of $y$ with respect to $x$ , we will use the power rule for differentiation. The power rule states that if $y = ax^n$ , then $\frac{dy}{dx} = n \cdot ax^{n-1}$ .

STEP 3

The term $4x^2$ can be differentiated using the power rule. Applying the power rule to this term gives us:
$\frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x$

STEP 4

The term $2\sqrt{x}$ can be rewritten as $2x^{1/2}$ to make it easier to differentiate using the power rule.

STEP 5

Now, apply the power rule to $2x^{1/2}$ :
$\frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{1/2 - 1} = x^{-1/2}$

STEP 6

Simplify the derivative of the second term:
$x^{-1/2} = \frac{1}{\sqrt{x}}$

STEP 7

Combine the derivatives of both terms to get the derivative of $y$ :
$\frac{dy}{dx} = 8x + \frac{1}{\sqrt{x}}$

STEP 8

Now, we need to evaluate $\frac{dy}{dx}$ at $x = 1$ . Substitute $x = 1$ into the derivative.
$\frac{dy}{dx}\bigg|_{x=1} = 8(1) + \frac{1}{\sqrt{1}}$

STEP 9

Simplify the expression by evaluating the square root of 1 and the multiplication:
$\frac{dy}{dx}\bigg|_{x=1} = 8 + 1$

STEP 10

Add the two numbers to find the value of the derivative at $x = 1$ :
$\frac{dy}{dx}\bigg|_{x=1} = 9$
The derivative of $y$ with respect to $x$ when $x = 1$ is 9.
The correct answer is: 9

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Find the derivative of $y=4x^2+2\sqrt{x}$ evaluated at $x=1$ .

STEP 1

Assumptions
1. The given function is $y = 4x^2 + 2\sqrt{x}$ .
2. We need to find the derivative of $y$ with respect to $x$ , denoted as $\frac{dy}{dx}$ .
3. After finding the derivative, we will evaluate $\frac{dy}{dx}$ at $x = 1$ .

STEP 2

To find the derivative of $y$ with respect to $x$ , we will use the power rule for differentiation. The power rule states that if $y = ax^n$ , then $\frac{dy}{dx} = n \cdot ax^{n-1}$ .

STEP 3

The term $4x^2$ can be differentiated using the power rule. Applying the power rule to this term gives us:
$\frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x$

STEP 4

The term $2\sqrt{x}$ can be rewritten as $2x^{1/2}$ to make it easier to differentiate using the power rule.

STEP 5

Now, apply the power rule to $2x^{1/2}$ :
$\frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{1/2 - 1} = x^{-1/2}$

STEP 6

Simplify the derivative of the second term:
$x^{-1/2} = \frac{1}{\sqrt{x}}$

STEP 7

Combine the derivatives of both terms to get the derivative of $y$ :
$\frac{dy}{dx} = 8x + \frac{1}{\sqrt{x}}$

STEP 8

Now, we need to evaluate $\frac{dy}{dx}$ at $x = 1$ . Substitute $x = 1$ into the derivative.
$\frac{dy}{dx}\bigg|_{x=1} = 8(1) + \frac{1}{\sqrt{1}}$

STEP 9

Simplify the expression by evaluating the square root of 1 and the multiplication:
$\frac{dy}{dx}\bigg|_{x=1} = 8 + 1$

STEP 10

Add the two numbers to find the value of the derivative at $x = 1$ :
$\frac{dy}{dx}\bigg|_{x=1} = 9$
The derivative of $y$ with respect to $x$ when $x = 1$ is 9.
The correct answer is: 9

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Find the derivative of y=4x2+2xy=4x^2+2\sqrt{x}y=4x2+2x​ evaluated at x=1x=1x=1.

STEP 1

Assumptions1. The given function is y=4x2+2x y = 4x^2 + 2\sqrt{x} y=4x2+2x​.2. We need to find the derivative of y y y with respect to x x x, denoted as dydx \frac{dy}{dx} dxdy​.3. After finding the derivative, we will evaluate dydx \frac{dy}{dx} dxdy​ at x=1 x = 1 x=1.

STEP 2

To find the derivative of y y y with respect to x x x, we will use the power rule for differentiation. The power rule states that if y=axn y = ax^n y=axn, then dydx=n⋅axn−1 \frac{dy}{dx} = n \cdot ax^{n-1} dxdy​=n⋅axn−1.

STEP 3

The term 4x2 4x^2 4x2 can be differentiated using the power rule. Applying the power rule to this term gives us:ddx(4x2)=4⋅2x2−1=8x \frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x dxd​(4x2)=4⋅2x2−1=8x

STEP 4

The term 2x 2\sqrt{x} 2x​ can be rewritten as 2x1/2 2x^{1/2} 2x1/2 to make it easier to differentiate using the power rule.

STEP 5

Now, apply the power rule to 2x1/2 2x^{1/2} 2x1/2:ddx(2x1/2)=2⋅12x1/2−1=x−1/2 \frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{1/2 - 1} = x^{-1/2} dxd​(2x1/2)=2⋅21​x1/2−1=x−1/2

STEP 6

Simplify the derivative of the second term:x−1/2=1x x^{-1/2} = \frac{1}{\sqrt{x}} x−1/2=x​1​

STEP 7

Combine the derivatives of both terms to get the derivative of y y y:dydx=8x+1x \frac{dy}{dx} = 8x + \frac{1}{\sqrt{x}} dxdy​=8x+x​1​

STEP 8

Now, we need to evaluate dydx \frac{dy}{dx} dxdy​ at x=1 x = 1 x=1. Substitute x=1 x = 1 x=1 into the derivative.dydx∣x=1=8(1)+11 \frac{dy}{dx}\bigg|_{x=1} = 8(1) + \frac{1}{\sqrt{1}} dxdy​∣∣​x=1​=8(1)+1​1​

STEP 9

Simplify the expression by evaluating the square root of 1 and the multiplication:dydx∣x=1=8+1 \frac{dy}{dx}\bigg|_{x=1} = 8 + 1 dxdy​∣∣​x=1​=8+1

STEP 10

Add the two numbers to find the value of the derivative at x=1 x = 1 x=1:dydx∣x=1=9 \frac{dy}{dx}\bigg|_{x=1} = 9 dxdy​∣∣​x=1​=9The derivative of y y y with respect to x x x when x=1 x = 1 x=1 is 9.The correct answer is: 9

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Find the derivative of $y=4x^2+2\sqrt{x}$ evaluated at $x=1$ .

Assumptions
1. The given function is $y = 4x^2 + 2\sqrt{x}$ .
2. We need to find the derivative of $y$ with respect to $x$ , denoted as $\frac{dy}{dx}$ .
3. After finding the derivative, we will evaluate $\frac{dy}{dx}$ at $x = 1$ .

To find the derivative of $y$ with respect to $x$ , we will use the power rule for differentiation. The power rule states that if $y = ax^n$ , then $\frac{dy}{dx} = n \cdot ax^{n-1}$ .

The term $4x^2$ can be differentiated using the power rule. Applying the power rule to this term gives us:
$\frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x$

The term $2\sqrt{x}$ can be rewritten as $2x^{1/2}$ to make it easier to differentiate using the power rule.

Now, apply the power rule to $2x^{1/2}$ :
$\frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{1/2 - 1} = x^{-1/2}$

Simplify the derivative of the second term:
$x^{-1/2} = \frac{1}{\sqrt{x}}$

Combine the derivatives of both terms to get the derivative of $y$ :
$\frac{dy}{dx} = 8x + \frac{1}{\sqrt{x}}$

Now, we need to evaluate $\frac{dy}{dx}$ at $x = 1$ . Substitute $x = 1$ into the derivative.
$\frac{dy}{dx}\bigg|_{x=1} = 8(1) + \frac{1}{\sqrt{1}}$

Simplify the expression by evaluating the square root of 1 and the multiplication:
$\frac{dy}{dx}\bigg|_{x=1} = 8 + 1$

Add the two numbers to find the value of the derivative at $x = 1$ :
$\frac{dy}{dx}\bigg|_{x=1} = 9$
The derivative of $y$ with respect to $x$ when $x = 1$ is 9.
The correct answer is: 9