Exercise · ZW-PRODUCT-RULE-004 · ★★☆ Intermediate
Differentiate \(f(x)=x \ln{\left(x \right)}\).
Choose one option.
Show answer
- D. \(\ln{\left(x \right)} + 1\)
Explanation
With \(u=x\) and \(v=\ln{\left(x \right)}\): \(u'=1\), \(v'=\frac{1}{x}\). Then \((uv)'=u'v+uv'=\ln{\left(x \right)} + 1\).
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-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.