What is the derivative of 1/ln x?
Understand the Problem
The question is asking for the derivative of the function 1/ln(x). To solve this, we will apply the quotient rule of differentiation, since it involves a division of two functions.
Answer
$ -\frac{1}{x [\ln(x)]^2} $
Answer for screen readers
The final answer is $ -\frac{1}{x [\ln(x)]^2} $
Steps to Solve
- Rewrite the function for easier differentiation
Rewrite the function using a negative exponent:
$$ f(x) = \frac{1}{\ln(x)} = [\ln(x)]^{-1} $$
This makes it easier to apply the chain rule.
- Apply the chain rule
The chain rule states that the derivative of $[u(x)]^n$ is $n[u(x)]^{n-1} , u'(x)$. Here, $u(x) = \ln(x)$ and $n = -1$:
$$ \frac{d}{dx}[\ln(x)]^{-1} = -1 [\ln(x)]^{-2} \frac{d}{dx}[\ln(x)] $$
- Find the derivative of $\ln(x)$
The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$:
$$ \frac{d}{dx}[\ln(x)] = \frac{1}{x} $$
- Combine the results
Substitute the derivative of $\ln(x)$ back into the previous result:
$$ \frac{d}{dx}[\ln(x)]^{-1} = -1 [\ln(x)]^{-2} \cdot \frac{1}{x}$$
Simplify the expression:
$$ \frac{d}{dx}\left( \frac{1}{\ln(x)} \right) = -\frac{1}{x[\ln(x)]^2} $$
The final answer is $ -\frac{1}{x [\ln(x)]^2} $
More Information
This form allows us to differentiate a composition of functions more easily. The result shows that the rate of change of $ \frac{1}{\ln(x)} $ decreases as x increases.
Tips
A common mistake is to forget to apply the chain rule properly or to incorrectly differentiate $\ln(x)$. Always remember that the derivative of $\ln(x)$ is $\frac{1}{x}$.