Exercise · ZW-QUOTIENT-RULE-008 · ★★★ Advanced
Compute \(f'(1)\) if \(f(x)=\frac{x^{3}}{x+2}\).
Choose one option.
Show answer
- D. \(\frac{8}{9}\)
Explanation
With \(u=x^{3}\) and \(v=x+2\): \(f'(x)=\frac{3x^{2}(x+2)-x^{3}}{(x+2)^{2}}\), which at \(x=1\) gives \(\frac{9-1}{9}=\frac{8}{9}\).
Review the concept: Quotient rule
The derivative of a quotient \(\frac{f}{g}\), with \(g\ne0\), is \(\frac{f^{\prime}g-fg^{\prime}}{g^2}\).
\[\left(\frac{f}{g}\right)'=\frac{f'g-f\,g'}{g^{2}}\]
More practice problems: quotient rule
- ZW-QUOTIENT-RULE-001 In the quotient rule ((f)/(g))', what is the correct numerator?
- ZW-QUOTIENT-RULE-002 Differentiate f(x)=(x)/(x + 1).
- ZW-QUOTIENT-RULE-003 Differentiate f(x)=(sin x)/(x).
- ZW-QUOTIENT-RULE-004 Differentiate f(x)=(x^2 + 1)/(x - 1).
- ZW-QUOTIENT-RULE-005 Differentiate f(x)=(e^x)/(x).
- ZW-QUOTIENT-RULE-006 Differentiate f(x)=(x^2)/(x^2 + 1).
- ZW-QUOTIENT-RULE-007 According to the quotient rule, what appears in the denominator of ((f)/(g))'?
- ZW-QUOTIENT-RULE-009 Which differentiation rule is used to differentiate a quotient (f)/(g) of two functions?