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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

  1. 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.

  1. 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 $\frac{1}{x}$.

  1. 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 $e$.
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