Questions: Differentiate. f(x)=8 ln x f'(x)=

Transcript text: Differentiate. \[ \begin{array}{l} f(x)=8 \ln x \\ f^{\prime}(x)=\square \end{array} \] $\square$

Solution

Solution Steps

To differentiate the function $ f(x) = 8 \ln x $, we will use the chain rule. The derivative of $ \ln x $ is $ \frac{1}{x} $. Therefore, the derivative of $ 8 \ln x $ is $ 8 $ times the derivative of $ \ln x $.

Step 1: Differentiate the Function

We start with the function $ f(x) = 8 \ln x $. To find the derivative $ f'(x) $, we apply the derivative rule for logarithmic functions. The derivative of $ \ln x $ is $ \frac{1}{x} $.

Step 2: Apply the Derivative Rule

Using the derivative rule, we have: \[ f'(x) = 8 \cdot \frac{d}{dx}(\ln x) = 8 \cdot \frac{1}{x} \]

Step 3: Simplify the Expression

Thus, the derivative simplifies to: \[ f'(x) = \frac{8}{x} \]