Calculus I · Unit 2A · lesson

Chain Rule with Exponentials and Logarithms

Concept

Learning objectives

Differentiate $e^{u(x)}$ , $a^{u(x)}$ , and $\ln|u(x)|$ .

Exponential and Logarithmic Compositions

Explanation

Exponential and logarithmic layers preserve their inner expressions

For $e^{g(x)}$ , the outer derivative remains $e^{g(x)}$ , then the chain rule contributes $g'(x)$ . For $\ln(g(x))$ , the outer derivative is the reciprocal of the inner expression, then multiplication by $g'(x)$ completes the rate conversion.

Parentheses are essential. Writing $g'(x)/g(x)$ makes the whole inner function the denominator, which is exactly what the logarithmic derivative requires.

Explanation

Before the formulas

In Chain Rule with Exponentials and Logarithms, the phrase "outside to inside" is useful only when paired with structure. Keep the inner expression unchanged while differentiating the outer layer, then multiply by the derivative of that inner expression. If the inner expression is itself composite, continue inward.

A missing inner derivative is the signature chain-rule error. Check units or scaling to catch it. If an inner quantity changes three times as fast, the final output rate should reflect that factor of three. The chain rule is a rate-conversion law, not punctuation attached to parentheses.

Exponential and logarithmic compositions appear whenever a model transforms a changing quantity multiplicatively or by scale. Their derivatives follow compact patterns, but each pattern is still the chain rule:

\frac{d}{dx}e^{u(x)}=e^{u(x)}u'(x), \qquad \frac{d}{dx}\ln u(x)=\frac{u'(x)}{u(x)}.

For logarithms, the sign and domain of $u$ require attention. In many derivative problems $d[\ln|u|]/dx=u'/u$ works on intervals where $u\ne0$ , but the underlying interval must not cross a zero of $u$ .

\boxed{\frac{d}{dx}e^{u}=e^u u'}, \qquad \boxed{\frac{d}{dx}a^u=a^u\ln(a)u'},

\boxed{\frac{d}{dx}\ln|u|=\frac{u'}u}.

Guided walkthrough

An exponential of a quadratic

Differentiate

y=e^{x^2-4x}.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Guided walkthrough

A logarithm that simplifies the chain rule

Differentiate

f(x)=\ln(x^2+1).

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A general exponential composition

For $y=5^{\sin x}$ ,

y'=5^{\sin x}\ln5\cos x.

Common mistake

(\ln u)'\ne\frac1x.

The derivative is $u'/u$ . The simple rule $1/x$ applies only when the logarithm's input is exactly $x$ .

After the explanation

Use the section idea

Reading lens

Read nested functions from the outside inward, but multiply local response factors through every layer.

Mental model

A small input change passes through a sequence of machines; the total response multiplies the response at each stage.

Decision

List the layers, differentiate one layer at a time, and stop only when every input-dependent layer contributes.

Common trap

Differentiating the outside and leaving the inside unchanged without its derivative factor.

Check yourself

Can you annotate every factor in your derivative with the layer that produced it?

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