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The derivative of a quotient

Calculus, grade 11 (4 units) Lesson 6 of 20Learning

The derivative of a quotient

A reminder: the rules of differentiation

In grade 10 you learnt to differentiate polynomials and roots. u\displaystyle u and v\displaystyle v are functions of x\displaystyle x, and c\displaystyle c is a number:

(xn)′=nxn−1\displaystyle \left(x^{n}\right)'=nx^{n-1}
(c⋅u)′\displaystyle \left(c\cdot u\right)' =\displaystyle = c⋅u′\displaystyle c\cdot u'
(u+v)′\displaystyle \left(u+v\right)' =\displaystyle = u′+v′\displaystyle u'+v'
(u⋅v)′\displaystyle \left(u\cdot v\right)' =\displaystyle = u′⋅v+u⋅v′\displaystyle u'\cdot v+u\cdot v'
(u)′\displaystyle \left(\sqrt{u}\right)' =\displaystyle = u′2u\displaystyle \frac{u'}{2\sqrt{u}}
  • The derivative of a number is 0.
  • A composite function: differentiate the outer function, and multiply by the derivative of the inner one – ((2x+1)3)′=3(2x+1)2⋅2\displaystyle {\left((2x+1)^{3}\right)'}={3(2x+1)^{2}\cdot2}.
The derivative of a quotient

A rational function is a quotient of two polynomials, uv\displaystyle \frac{u}{v}. Its derivative is given by the quotient rule:

(uv)′\displaystyle \left(\frac{u}{v}\right)' =\displaystyle = u′⋅v−u⋅v′v2\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}
  • In the numerator – first u′⋅v\displaystyle u'\cdot v, then subtract u⋅v′\displaystyle u\cdot v': the order changes the sign.
  • In the denominator – v2\displaystyle v^{2}. Leave it squared, without expanding the brackets: that way you see at once that it is never negative.
  • The derivative is defined on the same domain as the function: where v=0\displaystyle {v}={0} there is no function and no derivative.
Example

f(x)=x+1x−2\displaystyle {f(x)}={\frac{x+1}{x-2}}.
In the example below we differentiate by the quotient rule, with u=x+1\displaystyle {u}={x+1} and v=x−2\displaystyle {v}={x-2}:

  • u′=1\displaystyle {u'}={1}, v′=1\displaystyle {v'}={1};
  • f′(x)=1⋅(x−2)−(x+1)⋅1(x−2)2=x−2−x−1(x−2)2=−3(x−2)2\displaystyle {f'(x)}={\frac{1\cdot(x-2)-(x+1)\cdot1}{(x-2)^{2}}}={\frac{x-2-x-1}{(x-2)^{2}}}={\frac{-3}{(x-2)^{2}}}.

The numerator −3\displaystyle -3 is negative and the denominator positive, so f′(x)<0\displaystyle f'(x)<0 on the whole domain: f\displaystyle f falls on both sides of x=2\displaystyle {x}={2}.

A number over a function

When the numerator is a number c\displaystyle c, its derivative is 0, and the quotient rule leaves:

(cv)′\displaystyle \left(\frac{c}{v}\right)' =\displaystyle = −c⋅v′v2\displaystyle -\frac{c\cdot v'}{v^{2}}
(cx)′\displaystyle \left(\frac{c}{x}\right)' =\displaystyle = −cx2\displaystyle -\frac{c}{x^{2}}

For example, (4x)′=−4x2\displaystyle {\left(\frac{4}{x}\right)'}={-\frac{4}{x^{2}}}, and (1x2+1)′=−2x(x2+1)2\displaystyle {\left(\frac{1}{x^{2}+1}\right)'}={-\frac{2x}{(x^{2}+1)^{2}}}.

A polynomial plus a fraction

In a sum, each term is differentiated on its own: the polynomial as usual, and the fraction by the quotient rule.

(x+4x)′=1−4x2\left(x+\frac{4}{x}\right)'=1-\frac{4}{x^{2}}(x2−3x−1)′=2x+3(x−1)2\left(x^{2}-\frac{3}{x-1}\right)'=2x+\frac{3}{(x-1)^{2}}
Exercise 1

Differentiate.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 2

          f(x)=x2x+2\displaystyle {f(x)}={\frac{x^{2}}{x+2}}.

          a

          Find f′(x)\displaystyle f'(x).

          f(x)=x2x+2\displaystyle f(x)=\frac{x^{2}}{x+2}

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            b

            Work out f′(1)\displaystyle f'(1).

            f(x)=x2x+2\displaystyle f(x)=\frac{x^{2}}{x+2}

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 3

              Differentiate, and simplify the numerator fully.

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                Exercise 4

                f(x)=x−3x+1\displaystyle {f(x)}={\frac{x-3}{x+1}}.
                Find f′(x)\displaystyle f'(x).

                f(x)=x−3x+1\displaystyle f(x)=\frac{x-3}{x+1}

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  Explain with f′(x)\displaystyle f'(x): why does f\displaystyle f rise on its whole domain?