Exercise · ZW-BASIC-DIFFERENTIATION-RULES-008 · ★★★ Advanced
Differentiate \(h(x)=\sqrt[3]{x^{2}}+\frac{4}{x}\).
Choose one option.
Show answer
- C. \(\frac{2}{3\sqrt[3]{x}} - \frac{4}{x^{2}}\)
Explanation
Rewrite \(h(x)=x^{2/3}+4x^{-1}\) and apply the power rule to each term: \(\frac{2}{3}x^{-1/3}-4x^{-2}\), which equals \(\frac{2}{3\sqrt[3]{x}}-\frac{4}{x^{2}}\).
Review the concept: Basic differentiation rules
The derivative of a constant is \(0\) and that of \(x^n\) is \(n\,x^{n-1}\). The derivative is also linear: the derivative of a sum is the sum of the derivatives and constants factor out.
\[\begin{gathered}\frac{d}{dx}x^{n}=nx^{n-1}\\ (cf+g)'=cf'+g'\end{gathered}\]
More practice problems: basic differentiation rules
- ZW-BASIC-DIFFERENTIATION-RULES-001 Differentiate f(x)=5 x^4 - 3 x^2 + 7 x - 2.
- ZW-BASIC-DIFFERENTIATION-RULES-002 Differentiate f(x)=4 x^3 + 2 x - 9.
- ZW-BASIC-DIFFERENTIATION-RULES-003 Differentiate f(x)=x^5 - 3 x^2 + 7.
- ZW-BASIC-DIFFERENTIATION-RULES-004 Differentiate f(x)=3 √x + (1)/(x^2).
- ZW-BASIC-DIFFERENTIATION-RULES-005 Differentiate f(x)=x^6.
- ZW-BASIC-DIFFERENTIATION-RULES-006 Differentiate f(x)=(7)/(x^3).
- ZW-BASIC-DIFFERENTIATION-RULES-007 Differentiate g(t)=π^2+6 t^2.
- ZW-BASIC-DIFFERENTIATION-RULES-009 Differentiate f(x)=-8.
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