# QUOTIENT RULE OF DIFFERENTIATION

Have you ever wondered how we differentiate fractions and division? We use the quotient rule.

The quotient rule states that the derivative of a quotient is the denominator times the derivative of the numerator minus the derivative of the denominator times the numerator, all divided by the square of the denominator.

Symbolically, the quotient rule is represented as:

$$(\frac{f}{g})'=\frac{f'g-g'f}{(g)^2}$$

Example 1

Differentiate $y=\frac{5x}{x^3+2}$

Solution

Recalled that:

$$(\frac{f}{g})'=\frac{f'g-g'f}{(g)^2}$$

Here, f=5x, f'=5, g=x³+2, g'=3x².

$$\frac{dy}{dx}=\frac{5(x^3+2)-3x^2(5x)}{(x^3+2)^2}$$

$$\frac{dy}{dx}=\frac{5x^3+10-15x^³}{(x^3+2)^2}$$

$$\frac{dy}{dx}=\frac{10+10x^3}{(x+3)^2}$$

Example 2

Compute the derivative of $f(x)=\frac{x}{x-2}$ using the quotient rule

Solution

Recalled that:

$$(\frac{f}{g})'=\frac{f'g-g'f}{(g)^2}$$

Here, f=x, f'=1, g=x-2, g'=1.

$$\frac{dy}{dx}=\frac{1(x-2)-1(x)}{(x-2)^2}$$

$$\frac{dy}{dx}=\frac{x-2-x}{(x-2)^2}$$

$$\frac{dy}{dx}=\frac{-2}{(x-2)^2}$$

Example 3

If $h(x)=\frac{3x^2+1}{e^x}$, Evaluate $h'(x)$

Solution

f=3x²+1, f'=6x, g=e^x, g'=e^x

$$h'(x)=\frac{6x(e^x)-e^x(3x^2+1)}{(e^x)^2}$$

$$h'(x)=\frac{6xe^x-3x^2e^x-ex}{(e^x)^2}$$

$$h'(x)=\frac{e^x(6x-3x^2-1)}{(e^x)^2}$$

$$h'(x)=\frac{6x-3x^2-1}{e^x}$$

$$h'(x)=\frac{-3x^2+6x-1}{e^x}$$

Example 4

Differentiate $y=\frac{1-\frac{1}{x}}{1+\frac{2}{x}}$

Solution

The above expression can also be written as:

$$\frac{1-x^{-1}}{1+2x^{-1}}$$

Here $f=1-x^{-1}$, $f'=x^{-2}$, $g=1+2x^{-1}$, $g'=-2x^{-2}$

$$\frac{dy}{dx}=\frac{x^{-2}(1+2x^{-1})-(-2x^{-2}(1-x^{-1}))}{(1+2x^{-1})^2}$$

$$\frac{dy}{dx}=\frac{3x^{-2}}{(1+2x^{-1})^2}$$

$$\frac{dy}{dx}=\frac{3x^{-2}}{(1+\frac{2}{x})(1+\frac{2}{x})}$$

$$\frac{dy}{dx}=\frac{3x^{-2}}{1+\frac{2}{x}+\frac{2}{x}+\frac{4}{x^2}}$$

Taking the L.C.M

$$\frac{dy}{dx}=\frac{3x^{-2}}{\frac{x^2+2x+2x+4}{x^2}}$$

$$\frac{dy}{dx}=\frac{3}{x^2}÷\frac{x^2+4x+4}{x^2}$$

$$\frac{dy}{dx}=\frac{3}{x^2}\times\frac{x^2}{x^2+4x+4}$$

$$\frac{dy}{dx}=\frac{3x}{x^2+4x+4}$$

To sum up, the quotient rule is useful for differentiating division. Next, We will be looking at the chain rule of differentiation