Derivatives · Definition

Quotient rule

calculusderivatives

Before you start you need to know

The quotient rule differentiates a fraction of two functions without simplifying it first.

Definition

The derivative of a quotient \(\frac{f}{g}\), with \(g\ne0\), is \(\frac{f^{\prime}g-fg^{\prime}}{g^2}\).

\[\left(\frac{f}{g}\right)'=\frac{f'g-f\,g'}{g^{2}}\]

For a quotient, \(\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}\). The order in the numerator matters: first the derivative of the top times the bottom, then subtract.

Guide example: with \(u=x^{2}+1\) and \(v=x-1\), \(\left(\frac{u}{v}\right)'=\frac{2x(x-1)-(x^{2}+1)}{(x-1)^{2}}=\frac{x^{2}-2x-1}{(x-1)^{2}}\).

Typical error: writing \(f'g+fg'\) in the numerator or dividing by \(g\) instead of \(g^{2}\). The minus sign and the order matter, so check both.

Think about it

Differentiate \(\frac{x}{x+1}\).

\(\frac{(x+1)-x}{(x+1)^2}=\frac{1}{(x+1)^2}\).

Exercises 10

Differentiate the following functions

  1. \(\displaystyle f(x)=\frac{x}{x + 1}\)
  2. \(\displaystyle f(x)=\frac{\sin {x}}{x}\)
  3. \(\displaystyle f(x)=\frac{x^{2} + 1}{x - 1}\)
  4. \(\displaystyle f(x)=\frac{e^{x}}{x}\)
  5. \(\displaystyle f(x)=\frac{x^{2}}{x^{2} + 1}\)

Problems

  1. In the quotient rule \(\left(\dfrac{f}{g}\right)'\), what is the correct numerator?
  2. According to the quotient rule, what appears in the denominator of \(\left(\frac{f}{g}\right)'\)?
  3. Compute \(f'(1)\) if \(f(x)=\frac{x^{3}}{x+2}\).
  4. Which differentiation rule is used to differentiate a quotient \(\frac{f}{g}\) of two functions?
  5. Compute \(f'(1)\) if \(f(x)=\frac{2x+1}{x^{2}}\).

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.