How to find the derivative of logarithmic functions?
Understand the Problem
The question is asking how to find the derivative of logarithmic functions, which involves applying the rules of differentiation specifically for logarithmic expressions.
Answer
The derivative of $f(x) = \ln(x)$ is $f'(x) = \frac{1}{x}$.
Answer for screen readers
The derivative of $f(x) = \ln(x)$ is $f'(x) = \frac{1}{x}$. For a more complex function like $f(x) = \ln(g(x))$, the derivative is $f'(x) = \frac{g'(x)}{g(x)}$.
Steps to Solve
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Identify the Logarithmic Function
First, you need to identify the function you want to differentiate. For example, if you want to find the derivative of $f(x) = \ln(x)$, you recognize the logarithmic part. -
Recall the Derivative Rule for Logarithmic Functions
The derivative of the natural logarithm function is given by the formula:
$$ \frac{d}{dx} \ln(x) = \frac{1}{x} $$
This is valid for $x > 0$. -
Apply the Chain Rule (if necessary)
If the logarithmic function involves another function, such as $f(x) = \ln(g(x))$, you need to apply the chain rule. The derivative in this case will be:
$$ \frac{d}{dx} \ln(g(x)) = \frac{g'(x)}{g(x)} $$
where $g'(x)$ is the derivative of $g(x)$. -
Calculate the Derivative
If $f(x) = \ln(g(x))$, differentiate $g(x)$ first, then plug that back into the chain rule formula. For example, if $g(x) = x^2$, then $g'(x) = 2x$, and the derivative becomes:
$$ f'(x) = \frac{2x}{x^2} = \frac{2}{x} $$ -
Simplify the Result
Always try to simplify your answer if possible. For the previous example, the derivative of $f(x) = \ln(x^2)$ simplifies to:
$$ f'(x) = \frac{2}{x} $$
The derivative of $f(x) = \ln(x)$ is $f'(x) = \frac{1}{x}$. For a more complex function like $f(x) = \ln(g(x))$, the derivative is $f'(x) = \frac{g'(x)}{g(x)}$.
More Information
The natural logarithm function, $\ln(x)$, plays a crucial role in calculus, especially in solving problems involving exponential growth and decay. Understanding its derivative is essential for solving various problems in higher mathematics.
Tips
- Forgetting the Chain Rule: When differentiating composite functions that include logarithms, forgetting to apply the chain rule can lead to incorrect derivatives.
- Not considering the domain: The logarithmic function only accepts positive inputs, so be cautious about the domain of $x$ when applying the derivative.
