Exercise · ZW-CHAIN-RULE-008 · ★★★ Advanced
Differentiate \(f(x)=\cos(\ln(x^{2}+1))\).
Choose one option.
Show answer
- B. \(-\frac{2 x}{x^{2}+1}\sin(\ln(x^{2}+1))\)
Explanation
The outer function is \(\cos\) and the inner one is \(\ln(x^{2}+1)\). Differentiating gives \(-\sin(\ln(x^{2}+1))\cdot\frac{2x}{x^{2}+1}\), with a minus sign and the factor \(2x\).
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
- ZW-CHAIN-RULE-001 To differentiate sin (x^2), which is the inner function?
- ZW-CHAIN-RULE-002 Differentiate f(x)=sin x^2.
- ZW-CHAIN-RULE-003 Differentiate f(x)=(3 x + 1)^5.
- ZW-CHAIN-RULE-004 Differentiate f(x)=e^x^2.
- ZW-CHAIN-RULE-005 Differentiate f(x)=ln (x^2 + 1 ).
- ZW-CHAIN-RULE-006 Differentiate f(x)=√1 - x^2.
- ZW-CHAIN-RULE-007 Differentiate f(x)=(x^2 + x)^3.
- ZW-CHAIN-RULE-009 What is the derivative of e^3x?