8.2.4 f'(x)/f(x)

When you have an expression of the form $\ \frac{f'(x)}{f(x)}$, where $\ f(x)$ is a differentiable function, you can recognize this as the derivative of the natural logarithm of $\ f(x)$. This is because:

$\\ \frac{d}{dx}[\ln|f(x)|] = \frac{f'(x)}{f(x)}$

Explanation:

• The Derivative of $\ \ln|f(x)| :$

  • By the chain rule, the derivative of $\ \ln|f(x)|$ with respect to $\ x$ is: $\\ \frac{d}{dx}[\ln|f(x)|] = \frac{1}{f(x)} \cdot f'(x) = \frac{f'(x)}{f(x)}$

  • This means that if you encounter $\ \frac{f'(x)}{f(x)}$, it is the derivative of $\ \ln|f(x)| .$

Integration of $\ \frac{f'(x)}{f(x)}$:

If you need to integrate $\ \frac{f'(x)}{f(x)}$with respect to $\ x$, the result will be:

$\\ \int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$

Summary: