Exercise · ZW-PRODUCT-RULE-001 · ★★☆ Intermediate
What is the mistake in this statement? "\((x^{2}\sin x)'=2x\cos x\)"
Choose one option.
Show answer
- C. It differentiated each factor separately and multiplied the derivatives
Explanation
The derivative of a product is not the product of the derivatives. The rule is \((fg)'=f'g+fg'\): here it gives \(2x\sin x+x^{2}\cos x\).
Review the concept: Product rule
Rule giving the derivative of a product of two differentiable functions: \((fg)^{\prime}=f^{\prime}g+fg^{\prime}\). It is not the product of the derivatives.
\[(f\,g)'=f'\,g+f\,g'\]
More practice problems: product rule
- ZW-PRODUCT-RULE-002 Differentiate f(x)=x^2 sin x.
- ZW-PRODUCT-RULE-003 Differentiate f(x)=(3 x + 1) e^x.
- ZW-PRODUCT-RULE-004 Differentiate f(x)=x ln (x ).
- ZW-PRODUCT-RULE-005 Differentiate f(x)=(x^2 + 1) cos x.
- ZW-PRODUCT-RULE-006 Differentiate f(x)=e^x sin x.
- ZW-PRODUCT-RULE-007 Differentiate f(x)=x^3 e^x.
- ZW-PRODUCT-RULE-008 Differentiate f(x)=(x + 2) cos x.
- ZW-PRODUCT-RULE-009 Differentiate f(x)=x sin x e^x by applying the product rule twice.