Exercise · ZW-CHAIN-RULE-005 · ★★☆ Intermediate
Differentiate \(f(x)=\ln{\left(x^{2} + 1 \right)}\).
Choose one option.
Show answer
- A. \(\frac{2 x}{x^{2} + 1}\)
Explanation
The outer function is \(F(u)=\ln{\left(u \right)}\) and the inner one is \(g(x)=x^{2} + 1\). Then \(F'(g(x))\,g'(x)=\frac{1}{x^{2} + 1}\cdot\left(2 x\right)\), which gives \(\frac{2 x}{x^{2} + 1}\).
Review the concept: Chain rule
The chain rule gives the derivative of a composition \(f(g(x))\): the outer derivative evaluated at the inner function, times the derivative of the inner function.
\[\frac{d}{dx}f(g(x))=f'(g(x))\,g'(x)\]
More practice problems: chain rule
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- ZW-CHAIN-RULE-008 Differentiate f(x)=cos (ln (x^2+1)).
- ZW-CHAIN-RULE-009 What is the derivative of e^3x?