How to differentiate natural log?
Understand the Problem
The question is asking for the method to differentiate the natural logarithm (ln) function. This is a calculus problem where we need to recall the derivative rules for logarithmic functions.
Answer
The derivative of the natural logarithm function $y = \ln(x)$ is $\frac{dy}{dx} = \frac{1}{x}$.
Answer for screen readers
The derivative of the natural logarithm function $y = \ln(x)$ with respect to $x$ is:
$$ \frac{dy}{dx} = \frac{1}{x} $$
Steps to Solve
- Identify the natural logarithm function
The natural logarithm function can be represented as $y = \ln(x)$. To differentiate this function, we will apply the rules of derivatives in calculus.
- Use the derivative rule for the natural logarithm
The derivative of the natural logarithm function with respect to $x$ is given by the formula:
$$ \frac{dy}{dx} = \frac{1}{x} $$
This means that if $y = \ln(x)$, then the derivative $y'$ is simply $\frac{1}{x}$.
- Apply the derivative to a specific value
If we want to find the derivative at a specific point, say $x = a$, we substitute $a$ into our derivative:
$$ \frac{dy}{dx} \bigg|_{x=a} = \frac{1}{a} $$
This gives us the slope of the function at that point.
The derivative of the natural logarithm function $y = \ln(x)$ with respect to $x$ is:
$$ \frac{dy}{dx} = \frac{1}{x} $$
More Information
The natural logarithm is a fundamental concept in calculus, particularly in growth processes and logarithmic scales. The derivative tells us how quickly the function $y = \ln(x)$ is changing at any given point $x$.
Tips
- Forgetting that the domain of $\ln(x)$ is $x > 0$. Students may mistakenly try to evaluate the logarithm or its derivative at non-positive values.
- Confusing the natural logarithm with logarithms of other bases. Remember that $y = \ln(x)$ specifically refers to logarithm base $e$.