Quotient Rule

Quotient Rule is a method for finding the derivative of a function that is the quotient of two other functions.It is a method used for differentiating problems where one function is divided by another, a function of the form:f(x)/g(x).

\frac{d}{dx}[\frac{f(x)}{f(x)}]= \frac{g(x)\frac{d}{dx}[f(x)]-f(x)\frac{d}{dx}[g(x)}{[g(x)]^2}

Quotient Rule Formula

The quotient rule formula is the formula used to find the differentiation of the function, which is expressed as the quotient function.

d/dx [u(x)/v(x)] = [v(x) × u'(x) - u(x) × v'(x)] / [v(x)]²

Where,

• u(x) is the first function which is a differentiable function, 
• u'(x) is the derivative of function u(x), 
• v(x) is the second function which is a differentiable function, and
• v'(x) is the derivative of the function v(x).

Quotient Rule Proof

We can derive the quotient rule using the following methods:

• Using Chain Rule
• Using Implicit Differentiation
• Using Derivative and Limit Properties

Derivation of Quotient Rule Using Chain Rule

To Prove: H'(x) = d/dx [f(x)/g(x)] = [f'(x) × g(x) - g(x) × g'(x)] / [g(x)]²

Given: H(x) = f(x)/g(x)

Proof:

H(x) = f(x)/g(x)
⇒ H(x) = f(x).g(x)^-1

Using Product Rule,
H'(x) = f(x). d/dx [g(x)^-1] + g(x)^-1. f'(x)

Applying the power rule,
H'(x) = f(x). (-1)[g(x)^-2.g'(x)] + g(x)^-1. f'(x)
⇒ H'(x) = - [f(x).g'(x)] / [g(x)]² + f'(x) / [g(x)]

H'(x) = [-f'(x).g'(x) - f'(x).g(x)] / [g(x)]^2

Thus, the quotient rule is proved.

Derivation of Quotient Rule Using Implicit Differentiation

Let's take a differentiable function f(x), such that f(x) = u(x)/v(x).

u(x) = f(x).v(x)

using the product rule,
u'(x) = f'(x)·v(x) + f(x)·v'(x)

Now solving for f'(x)
f'(x) = [u'(x) - f(x)·v'(x)] / v(x)

Substituting the value of f(x) as, f(x) = u(x)/v(x)
f'(x) = [u'(x) - (u(x)/v(x))·v'(x)] / v(x)

f'(x) = [u'(x)·v(x) - u(x)·v'(x)] / v²(x)

Thus, the quotient rule is proved.

Derivation of Quotient Rule Using Derivative and Limit Properties

Let's take a differentiable function f(x) such that f(x) = u(x)/v(x),

We know that, f'(x) = lim_h→0 [f(x+h) - f(x)] / h

Substituting the value of f(x) = u(x)/v(x)
f'(x) = lim_h→0 [u(x+h)/v(x+h) - u(x)/v(x)] / h
f'(x) = lim_h→0 [u(x+h).v(x) - u(x).v(x+h)] / h.v(x).v(x+h)

Distributing the limit,

f'(x) = {lim_h→0 [u(x+h).v(x) - u(x).v(x+h)] / h}.{lim_h→0 1/v(x).v(x+h)}
⇒ f'(x) = {lim_h→0 [u(x+h).v(x) - u(x).v(x+h) + u(x)v(x) - u(x)v(x)] / h}.{1/v(x).v(x)}
⇒ f'(x) = {lim_h→0 [u(x+h).v(x) - u(x).v(x)] / h} {lim_h→0  [u(x)v(x+h) - u(x)v(x)] / h}.{1/v²(x)}
⇒ f'(x) = v(x){lim_h→0 [u(x+h) - u(x)] / h} -u(x) {lim_h→0  [-v(x+h) + v(x)] / h}.{1/v²(x)}

f'(x) = [v(x).u'(x) - u(x).v'(x)] / v²(x)

Which is the required quotient rule.

How to Use Quotient Rule in Differentiation?

Step 1: Write the individual functions as u(x) and v(x).

Step 2: Find the derivative of the individual function u(x) and v(x), i.e. find u'(x) and v'(x). Now apply the quotient rule formula,

f'(x) = [u(x)/v(x)]' = [u'(x) × v(x) - u(x) × v'(x)] / [v(x)]²

Step 3: Simplify the above equation and it gives the differentiation of f(x).

We can understand this concept with the help of an example.

Example: Find f'(x) if f(x) = 2x³/(x+2)

Given, f(x) = 2x³/(x + 2)

Comparing with f(x) = u(x)/v(x), we get

• u(x) = 2x³
• v(x) = (x + 2)

Now Differentiating u(x) and v(x)

• u'(x) = 6x²
• v'(x) = 1

Using Quotient rule,

f'(x) = [v(x)u'(x) - u(x)v'(x)]/[v(x)]²
⇒ f'(x) = [(x+2)•6x² - 2x³•1]/(x + 2)²
⇒ f'(x) = (6x³ + 12x² - 2x³)/(x + 2)²
⇒ f'(x) = (4x³ + 12x²​​​​)/(x + 2)²

Product and Quotient Rule

The product rule of differentiation is used to find the differentiation of a function when the function is given as product of two function.

Product rule of differentiation states that , if P(x) = f(x).g(x)

P'(x) = f(x).g'(x) + f'(x).g(x)

Whereas the quotient rule of differentiation is used to differentiate a function that is represented as, division of two functions, i.e. f(x) = p(x)/q(x).

Then the derivation of f(x) using the quotient rule is calculated as,

f'(x) = {q(x).p'(x) - p(x).q'(x)}/q²(x)

Solved Examples on Quotient Rule

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

Here are a few practice problems on the Quotient Rule for you to solve.

Question 1: Find the derivative of f(x) = (x² + 3)/(sin x)

Question 2: Find the derivative of f(x) = (2x² + 3x + 5)/(x + 3)

Question 3: Find the derivative of f(x) = (x + 3)/(ln x)

Question 4: Find the derivative of f(x) = (x.sin x)/(x²)

Question 5: Differentiate f(x)= \frac{Sin(x)}{x^2}

Question 6: Find the derivative of f(x) = \frac{Cos(x)}{1+Sin(x)}

Question 7: Use the quotient rule to differentiate f(x) = \frac{x^2+3x+2}{x-4}

Question 8: Compute the derivative of f(x)= \frac{2x+3}{4x^2-5}

Question 9: Differentiate f(x)= \frac{x^3-2x}{√x}

Question 10: Find the derivative of f(x) = \frac{\tan{x}}{x}

Related Reads

• Product Rule in Calculus
• Chain Rule
• Differentiation and Integration Formula
• Logarithmic Differentiation
• Fundamentals of Calculus
• Application of Derivatives
• Implicit Differentiation
• Properties of Limits
• Rules of Derivatives