Math / Calculus

QuestionFind the derivative of the function $F(x)=\ln \left(\frac{x}{4}\right)$ .

Studdy Solution

STEP 1

1. We are given the function $F(x) = \ln \left(\frac{x}{4}\right)$ .
2. We need to find the derivative of this function with respect to $x$ .

STEP 2

1. Use the properties of logarithms to simplify the function.
2. Differentiate the simplified function with respect to $x$ .

STEP 3

Use the property of logarithms that states $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$ to simplify the function:
$F(x) = \ln(x) - \ln(4)$

STEP 4

Differentiate the simplified function with respect to $x$ . Recall that the derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$ , and the derivative of a constant is $0$ :
$F'(x) = \frac{d}{dx}[\ln(x) - \ln(4)]$ $F'(x) = \frac{d}{dx}[\ln(x)] - \frac{d}{dx}[\ln(4)]$ $F'(x) = \frac{1}{x} - 0$ $F'(x) = \frac{1}{x}$
The derivative of the function $F(x) = \ln \left(\frac{x}{4}\right)$ is:
$\boxed{\frac{1}{x}}$

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