derivative of ln absolute value of x
Understand the Problem
The question is asking for the derivative of the natural logarithm of the absolute value of x, which involves applying the rules of differentiation. We will use the chain rule and properties of logarithms to find the solution.
Answer
The derivative is $f'(x) = \frac{1}{x} \text{ for } x \neq 0$.
Answer for screen readers
The derivative of the natural logarithm of the absolute value of $x$ is: $$ f'(x) = \frac{1}{x} \text{ for } x \neq 0 $$
Steps to Solve
- Start with the function
We are given the function to differentiate: $$ f(x) = \ln(|x|) $$
- Apply the chain rule
To find the derivative $f'(x)$, we apply the chain rule. The chain rule states that if we have a composite function, its derivative is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.
Here, the outer function is $\ln(u)$ where $u = |x|$.
The derivative of $\ln(u)$ is: $$ f'(x) = \frac{1}{|x|} \cdot \frac{d}{dx}(|x|) $$
- Differentiate the absolute value
Next, we find the derivative of $|x|$.
The derivative of $|x|$ is: $$ \frac{d}{dx}(|x|) = \begin{cases} 1 & \text{if } x > 0 \ -1 & \text{if } x < 0 \ 0 & \text{if } x = 0 \end{cases} $$
- Combine results for the derivative
Now we can combine these results. Since the function $|x|$ is positive when $x \neq 0$, we can express the derivative as: $$ f'(x) = \begin{cases} \frac{1}{x} & \text{if } x > 0 \ \frac{-1}{x} & \text{if } x < 0 \end{cases} $$
- Final derivative expression
Thus, the derivative of the natural logarithm of the absolute value of $x$ is: $$ f'(x) = \frac{1}{x} \text{ for } x \neq 0 $$
The derivative of the natural logarithm of the absolute value of $x$ is: $$ f'(x) = \frac{1}{x} \text{ for } x \neq 0 $$
More Information
The derivative $f'(x) = \frac{1}{x}$ shows that the rate of change of the function $\ln(|x|)$ depends on the sign of $x$. This result is significant in calculus and is used in various applications, especially in integral calculations and optimization problems.
Tips
One common mistake is forgetting to consider the case when $x = 0$. The absolute value function is not differentiable at zero, so one should clearly note that $f'(x)$ is defined for $x \neq 0$.
