Practice Question on Quotient Rule

The Quotient Rule is a differentiation rule used to determine the derivative of a function that is expressed as the quotient (division) of two differentiable functions.

Solved Examples

Example 1: Differentiate \bold{y=\frac{x^3-x+2}{x^2+5}}   .

Solution :

Both Numerator and Denominator functions are differentiable.

Applying Quotient Rule,

y'=\dfrac {d}{dx}[\dfrac{x^3-5+2}{x^2+5}]

 y'= \dfrac{[d/dx(x^3-x+2)(x^2+5)-(x^3-x+2)d/dx(x^2+5)]}{[x^2+5]^2}

 y'= \dfrac{[(3x^2-1)(x^2+5)-(x^3-x+2)(2x)]}{[x^2+5]^2}\\=\dfrac{(3x^4+15x^2-x^2-5)-(2x^4-2x^2+4x)}{[x^2+5]^2}

 y'= \dfrac{x^4+16x^2-4x-5}{[x^2+5]^2}

Example 2: Differentiate, f(x) = tan x.

Solution :

tan x is written as sinx/cosx, i.e.

tan x = (sin x) / (cos x)

Both Numerator and Denominator functions are differentiable.

Applying Quotient Rule,

f'(x)=\dfrac{(d/dx(sinx))(cosx)-(d/dx(cosx))(sinx)}{cos^2x}

⇒ f'(x)= \dfrac{cosx.cosx-(-sinx)(sinx)}{cos^x}

⇒ f'(x)=\dfrac{cos^2x+sin^2x}{cos^2x}

⇒ f'(x)=\dfrac{1}{cos^2x}=sec^2x.

Example 3: Differentiate, f(x)= e^x/x²

Solution :

Both Numerator and Denominator functions are differentiable.

Applying Quotient Rule,

f'(x)=\dfrac{[(x^2)\frac{d}{dx} (e^x)-(e^x)\frac{d}{dx}(x^2)]}{x^4}

Differentiate:

\frac{d}{dx}(e^x) = e^x,\ \frac{d}{dx}(x^2) = 2x

Substitute and simplify:

f'(x)=\frac{x^2e^x-2xe^x}{x^4} \\f'(x)=\frac{e^x(x^2-2x)}{x^4} \\f'(x)=\frac{e^x(x-2)}{x^3}

⇒ f'(x)=\frac{e^x(x-2)}{x^3} a

Example 4: Differentiate, y=\frac{cosx}{x^2}

Solution :

Both Numerator and Denominator functions are differentiable.

Applying Quotient Rule,

y'=\dfrac{d/dx(cosx)(x^2)-d/dx(x^2)(cosx)}{x^4}

⇒ y'=\dfrac{-sinx(x^2)-(2x)(cosx)}{x^4}

⇒ y'=\dfrac{-(x^2)sinx-(2xcosx)}{x^4}

Example 5: Differentiate, f(p) = p+5/p+7

Solution :

Both Numerator and Denominator functions are differentiable.

Applying Quotient Rule,

f'(p)=d/dx[\dfrac{p+5}{p+7}]

⇒ f'(p)=[\dfrac{d/dx(p+5)(p+7)-d/dx(p+7)(p+5)}{(p+7)^2}]

⇒ f'(p)=[\dfrac{p+7-p-5}{(p+7)^2}]

⇒ f'(p)=[\dfrac{2}{(p+7)^2}]

Practice Problems

Q1: Find the derivative of f(x) = (x² + 3)/(sin x)
Q2: Find the derivative of f(x) = (2x² + 3x + 5)/(x + 3)
Q3: Find the derivative of f(x) = (x + 3)/(ln x)
Q4: Find the derivative of f(x) = (x.sin x)/(x²)
Q5: Differentiate f(x)= \frac{Sin(x)}{x^2}

Answer :
1.f'(x)=\frac{2x\sin x-(x^2+3)\cos x}{\sin^2 x}

2. f'(x)=\frac{2x^2+12x+4}{(x+3)^2}

3. f'(x)=\frac{\ln x-\frac{x+3}{x}}{(\ln x)^2}

4. f'(x)=\frac{x\cos x-\sin x}{x^2}

5. f'(x)=\frac{x\cos x-2\sin x}{x^3}