Solved on Jan 13, 2024

Find the derivative of $\ln \left(\frac{x}{2}\right)$ .

STEP 1

Assumptions
1. We are given the function $f(x) = \ln\left(\frac{x}{2}\right)$ .
2. We need to find the derivative of the function with respect to $x$ .
3. We will use the chain rule and the properties of logarithms to find the derivative.

STEP 2

Recall the chain rule for differentiation, which states that if you have a composite function $g(f(x))$ , then its derivative is $g'(f(x)) \cdot f'(x)$ .

STEP 3

Recognize that the function $f(x) = \ln\left(\frac{x}{2}\right)$ can be rewritten using the property of logarithms that $\ln(a/b) = \ln(a) - \ln(b)$ .

STEP 4

Apply the property of logarithms to rewrite the function.
$f(x) = \ln\left(\frac{x}{2}\right) = \ln(x) - \ln(2)$

STEP 5

Notice that $\ln(2)$ is a constant, and the derivative of a constant is zero.

STEP 6

Now, differentiate the function with respect to $x$ using the chain rule.
$\frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{d}{dx}\ln(x) - \frac{d}{dx}\ln(2)$

STEP 7

Calculate the derivative of $\ln(x)$ with respect to $x$ .
$\frac{d}{dx}\ln(x) = \frac{1}{x}$

STEP 8

Calculate the derivative of $\ln(2)$ with respect to $x$ .
$\frac{d}{dx}\ln(2) = 0$

STEP 9

Combine the results from STEP_7 and STEP_8.
$\frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{1}{x} - 0$

STEP 10

Simplify the expression.
$\frac{d}{dx}\left(\ln\left(\frac{x}{2}\right)\right) = \frac{1}{x}$
The correct option is $\frac{1}{x}$ .

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Find the derivative of $\ln \left(\frac{x}{2}\right)$ .

STEP 1

Assumptions
1. We are given the function $f(x) = \ln\left(\frac{x}{2}\right)$ .
2. We need to find the derivative of the function with respect to $x$ .
3. We will use the chain rule and the properties of logarithms to find the derivative.

STEP 2

Recall the chain rule for differentiation, which states that if you have a composite function $g(f(x))$ , then its derivative is $g'(f(x)) \cdot f'(x)$ .

STEP 3

Recognize that the function $f(x) = \ln\left(\frac{x}{2}\right)$ can be rewritten using the property of logarithms that $\ln(a/b) = \ln(a) - \ln(b)$ .

STEP 4

Apply the property of logarithms to rewrite the function.
$f(x) = \ln\left(\frac{x}{2}\right) = \ln(x) - \ln(2)$

STEP 5

Notice that $\ln(2)$ is a constant, and the derivative of a constant is zero.

STEP 6

Now, differentiate the function with respect to $x$ using the chain rule.
$\frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{d}{dx}\ln(x) - \frac{d}{dx}\ln(2)$

STEP 7

Calculate the derivative of $\ln(x)$ with respect to $x$ .
$\frac{d}{dx}\ln(x) = \frac{1}{x}$

STEP 8

Calculate the derivative of $\ln(2)$ with respect to $x$ .
$\frac{d}{dx}\ln(2) = 0$

STEP 9

Combine the results from STEP_7 and STEP_8.
$\frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{1}{x} - 0$

STEP 10

Simplify the expression.
$\frac{d}{dx}\left(\ln\left(\frac{x}{2}\right)\right) = \frac{1}{x}$
The correct option is $\frac{1}{x}$ .

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Find the derivative of ln⁡(x2)\ln \left(\frac{x}{2}\right)ln(2x​).

STEP 1

Assumptions1. We are given the function f(x)=ln⁡(x2) f(x) = \ln\left(\frac{x}{2}\right) f(x)=ln(2x​).2. We need to find the derivative of the function with respect to x x x.3. We will use the chain rule and the properties of logarithms to find the derivative.

STEP 2

Recall the chain rule for differentiation, which states that if you have a composite function g(f(x)) g(f(x)) g(f(x)), then its derivative is g′(f(x))⋅f′(x) g'(f(x)) \cdot f'(x) g′(f(x))⋅f′(x).

STEP 3

Recognize that the function f(x)=ln⁡(x2) f(x) = \ln\left(\frac{x}{2}\right) f(x)=ln(2x​) can be rewritten using the property of logarithms that ln⁡(a/b)=ln⁡(a)−ln⁡(b) \ln(a/b) = \ln(a) - \ln(b) ln(a/b)=ln(a)−ln(b).

STEP 4

Apply the property of logarithms to rewrite the function.f(x)=ln⁡(x2)=ln⁡(x)−ln⁡(2) f(x) = \ln\left(\frac{x}{2}\right) = \ln(x) - \ln(2) f(x)=ln(2x​)=ln(x)−ln(2)

STEP 5

Notice that ln⁡(2) \ln(2) ln(2) is a constant, and the derivative of a constant is zero.

STEP 6

Now, differentiate the function with respect to x x x using the chain rule.ddx(ln⁡(x)−ln⁡(2))=ddxln⁡(x)−ddxln⁡(2) \frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{d}{dx}\ln(x) - \frac{d}{dx}\ln(2) dxd​(ln(x)−ln(2))=dxd​ln(x)−dxd​ln(2)

STEP 7

Calculate the derivative of ln⁡(x) \ln(x) ln(x) with respect to x x x.ddxln⁡(x)=1x \frac{d}{dx}\ln(x) = \frac{1}{x} dxd​ln(x)=x1​

STEP 8

Calculate the derivative of ln⁡(2) \ln(2) ln(2) with respect to x x x.ddxln⁡(2)=0 \frac{d}{dx}\ln(2) = 0 dxd​ln(2)=0

STEP 9

Combine the results from STEP_7 and STEP_8.ddx(ln⁡(x)−ln⁡(2))=1x−0 \frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{1}{x} - 0 dxd​(ln(x)−ln(2))=x1​−0

STEP 10

Simplify the expression.ddx(ln⁡(x2))=1x \frac{d}{dx}\left(\ln\left(\frac{x}{2}\right)\right) = \frac{1}{x} dxd​(ln(2x​))=x1​The correct option is 1x \frac{1}{x} x1​.

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Find the derivative of $\ln \left(\frac{x}{2}\right)$ .

Assumptions
1. We are given the function $f(x) = \ln\left(\frac{x}{2}\right)$ .
2. We need to find the derivative of the function with respect to $x$ .
3. We will use the chain rule and the properties of logarithms to find the derivative.

Recall the chain rule for differentiation, which states that if you have a composite function $g(f(x))$ , then its derivative is $g'(f(x)) \cdot f'(x)$ .

Recognize that the function $f(x) = \ln\left(\frac{x}{2}\right)$ can be rewritten using the property of logarithms that $\ln(a/b) = \ln(a) - \ln(b)$ .

Apply the property of logarithms to rewrite the function.
$f(x) = \ln\left(\frac{x}{2}\right) = \ln(x) - \ln(2)$

Notice that $\ln(2)$ is a constant, and the derivative of a constant is zero.

Now, differentiate the function with respect to $x$ using the chain rule.
$\frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{d}{dx}\ln(x) - \frac{d}{dx}\ln(2)$

Calculate the derivative of $\ln(x)$ with respect to $x$ .
$\frac{d}{dx}\ln(x) = \frac{1}{x}$

Calculate the derivative of $\ln(2)$ with respect to $x$ .
$\frac{d}{dx}\ln(2) = 0$

Combine the results from STEP_7 and STEP_8.
$\frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{1}{x} - 0$

Simplify the expression.
$\frac{d}{dx}\left(\ln\left(\frac{x}{2}\right)\right) = \frac{1}{x}$
The correct option is $\frac{1}{x}$ .