Exercise · ZW-PRODUCT-RULE-009 · ★★★ Advanced
Differentiate \(f(x)=x \sin {x}\, e^{x}\) by applying the product rule twice.
Choose one option.
Show answer
- C. \(e^{x}\left(\sin {x} + x \cos {x} + x \sin {x}\right)\)
Explanation
With \(u=x \sin {x}\) and \(v=e^{x}\): \(u'=\sin {x}+x \cos {x}\), \(v'=e^{x}\). Then \(f'=u'v+uv'=e^{x}\left(\sin {x}+x \cos {x}+x \sin {x}\right)\).
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-001 What is the mistake in this statement? "(x^2sin x)'=2xcos x"
- 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.