Exercise · ZW-CHAIN-RULE-002 · ★☆☆ Basic
Differentiate \(f(x)=\sin {x^{2}}\).
Choose one option.
Show answer
- D. \(2 x \cos {x^{2}}\)
Explanation
The outer function is \(F(u)=\sin {u}\) and the inner one is \(g(x)=x^{2}\). Then \(F'(g(x))\,g'(x)=\cos {x^{2}}\cdot\left(2 x\right)\), which gives \(2 x \cos {x^{2}}\).
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-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-008 Differentiate f(x)=cos (ln (x^2+1)).
- ZW-CHAIN-RULE-009 What is the derivative of e^3x?