Logarithmic Function Differentiation

Logarithmic Function Differentiation

When differentiating a logarithmic function, the key idea is to use the rule for the derivative of (\ln(x)) and combine it with the chain rule when the input is not just (x).

1) Identify the logarithm

Look for expressions such as (\ln(x)), (\ln(g(x))), or other logarithmic forms built from a function inside the logarithm. The inside function matters because it changes the derivative.

2) Differentiate the outside and the inside

• For (\ln(x)), the derivative is (1/x).
• For (\ln(g(x))), apply the chain rule: [ \frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}. ] This means: keep the denominator as the original inside function, then multiply by the derivative of that inside function.

3) Simplify carefully

After differentiating, reduce fractions and combine terms if needed. Write your final answer in the simplest form possible.

4) Check your result

A good check is to see whether the derivative has the expected structure: if the original function is a logarithm, the result should usually be a fraction involving the inside function. Also verify that any chain rule factor is included.

Common mistake to avoid

Do not differentiate (\ln(g(x))) as if it were just (\ln(x)). The derivative of the inside function must appear.