Exercise · ZW-QUOTIENT-RULE-006 · ★★☆ Intermediate
Differentiate \(f(x)=\frac{x^{2}}{x^{2} + 1}\).
Choose one option.
Show answer
- B. \(\frac{2 x}{\left(x^{2} + 1\right)^{2}}\)
Explanation
With \(u=x^{2}\) and \(v=x^{2} + 1\): \(u'=2 x\), \(v'=2 x\). Then \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^{2}}=\frac{2 x}{\left(x^{2} + 1\right)^{2}}\).
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-005 Differentiate f(x)=(e^x)/(x).
- 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?