Solved on Jan 13, 2024

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

STEP 1

Assumptions
1. We are given the function f(x)=ln(x2) f(x) = \ln\left(\frac{x}{2}\right) .
2. We need to find the derivative of the function with respect to 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)) , then its derivative is g(f(x))f(x) g'(f(x)) \cdot f'(x) .

STEP 3

Recognize that the function f(x)=ln(x2) f(x) = \ln\left(\frac{x}{2}\right) can be rewritten using the property of logarithms that 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)

STEP 5

Notice that 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 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)

STEP 7

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

STEP 8

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

STEP 9

Combine the results from STEP_7 and STEP_8.
ddx(ln(x)ln(2))=1x0 \frac{d}{dx}\left(\ln(x) - \ln(2)\right) = \frac{1}{x} - 0

STEP 10

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

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