Exercise · ZW-QUOTIENT-RULE-005 · ★★☆ Intermediate
Differentiate \(f(x)=\frac{e^{x}}{x}\).
Choose one option.
Show answer
- C. \(\frac{x e^{x} - e^{x}}{x^{2}}\)
Explanation
With \(u=e^{x}\) and \(v=x\): \(u'=e^{x}\), \(v'=1\). Then \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^{2}}=\frac{x e^{x} - e^{x}}{x^{2}}\).
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-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-008 Compute f'(1) if f(x)=(x^3)/(x+2).
- ZW-QUOTIENT-RULE-009 Which differentiation rule is used to differentiate a quotient (f)/(g) of two functions?