How to Use Quotient Rule?

Quotient rule is an important of derivatives. To find the derivatives of complex fractions this quotient rule is used. The quotient rule helps to find the derivative of complex fractions very easily. It is used to find the derivative when the problem is given in fraction form i.e. in the numerator and denominator form.

This article is about the quotient rule in derivatives, and how it is applied and used.

Quotient rule is the rule for finding the derivative of the functions which is in the fraction form of two differentiable functions.

Quotient rule states that the derivative of the quotient is equal to the ratio of subtraction of the denominator function multiplied by the derivative of the numerator function and the numerator function multiplied by the derivative of the denominator function to the square of the denominator.

Quotient rule is applied when we have to find the derivative of a function of the form first function divided by the second function, which says that the derivative of the quotient is equal to the subtraction of the second function multiplied by the derivative of the first function and first function multiplied by the derivative of the second function divided by the square of the second function.

Quotient Rule is a method in calculus for finding the derivative of a function that is the ratio of two differentiable functions. The quotient rule formula is given below:

d/dx{g(x)/h(x)} = [g(x) × h'(x) – h(x) × g'(x)] / [h(x)]²

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
• v'(x) is the derivative of the function v(x)

To use Quotient Rule follow the steps added below:

Step 1: Identify the Functions u(x) and v(x)

Step 2: Differentiate u(x) to get u′(x)

Step 3: Differentiate v(x) to get v′(x)

Step 41: Apply the Quotient Rule Formula (as added above)

Step 5: Simplify the Result

Related examples are added below:

Example 1: Find  [Tex]\dfrac{d}{dx}\left[\dfrac{3x^2+ 6x+2}{x+2}\right][/Tex].

Solution:

 [Tex]\dfrac{d}{dx}[\dfrac{3x^2+ 6x+2}{x+2}] = \dfrac{(x+2)\dfrac{d}{dx}(3x^2+6x+2)-(3x^2+6x+2)\dfrac{d}{dx}(x+2)}{(x+2)^2} [/Tex]

= [(x + 2)(6x+6) – (3x²+6x+2)(1)]/(x + 2)²

= [(6x²+6x+12x+12) – (3x²+6x+2)]/(x + 2)²

= (3x²+12x+10)/(x + 2)²

Example 2: Find [Tex]\dfrac{d}{dx}[\dfrac{sin x}{x^2}][/Tex]

Solution:

[Tex]\dfrac{d}{dx}[\dfrac{sin x}{x^2}] =  \dfrac{x^2.\dfrac{d}{dx}[sin x] – sin x.\dfrac{d}{dx}[x^2] }{[x^2]^2}  [/Tex] 

= (x²cosx – 2xsinx)/x⁴ 

= x.(xcosx – 2sinx)/x⁴

= (xcosx – 2sinx)/x³ 

Example 3: Find [Tex]\dfrac{d}{dx}[\dfrac{cos(x^2) }{x}]       [/Tex]      

Solution:

[Tex]\dfrac{d}{dx}[\dfrac{cos (x^2)}{x}] =  \dfrac{x.\dfrac{d}{dx}[cos(x^2 )] – cos(x^2) .\dfrac{d}{dx}[x] }{(x)^2}      [/Tex] 

= [-2x²sin(x²) – cos(x²)]/x² 

Example 4: Find [Tex]\dfrac{d}{dx}[\dfrac{2x}{5x^2+x+7}]  [/Tex]

Solution:

[Tex]\dfrac{d}{dx}[\dfrac{2x}{5x^2+x+7}] = \dfrac{(5x^2+x+7).\dfrac{d}{dx}(2x)-(2x).\dfrac{d}{dx}(5x^2+x+7)}{[5x^2+x+7]^2} [/Tex]

= [(5x²+x+7)(2) – (2x)(10x+1)]/(5x²+x+7)²

= [(10x²+2x+14)-(20x²+2x)]/(5x²+x+7)²

= [14-10x²)]/(5x²+x+7)²

Example 5: Find the derivative of  y = (e^x + log x)/sin3x

Solution:

By quotient rule,

[Tex]\dfrac{d}{dx}[\dfrac{e^x + log x}{sin3x}] = \dfrac{sin3x.\dfrac{d}{dx}[e^x + log x]-(e^x + log x).\dfrac{d}{dx}[sin3x]}{(sin3x)^2}   [/Tex]          

= [sin3x . {e^x + (1/x)} – (e^x + log x)(3cos3x)]/sin²3x

= [{e^x + (1/x)}.sin3x – 3(e^x + log x)cos3x ]/sin²3x

Example 6: Find the derivative of  y = (e^x + e^-x)/ (e^x – e^-x).

Solution:

[Tex]\dfrac{dy}{dx} = \dfrac{d}{dx}[\dfrac{(e^x + e^{-x}) }{(e^x – e^{-x})}] [/Tex]

=[Tex] \dfrac{d}{dx}[\dfrac{(e^x – e^{-x}).\dfrac{d}{dx}(e^x + e^{-x})-(e^x + e^{-x}).\dfrac{d}{dx}(e^x – e^{-x})  }{(e^x – e^{-x})^2}]  [/Tex] 

= [(e^x – e^-x)(e^x – e^-x) – (e^x + e^-x)(e^x + e^-x)]/(e^x – e^-x)² 

= [(e^x – e^-x)² – (e^x + e^-x)²]/(e^x – e^-x)²

= 4/(e^x – e^-x)² 

Q1. Differentiate the following function using the quotient rule: f(x) = (x² + 3x)/(x − 2)

Q2. Find the derivative of the function: g(x) = (2x + 5)/(x² – 1)

Q3. Use the quotient rule to differentiate: h(x) = (x³ – x + 1)/(2x + 3)

Q4. Compute the derivative of: k(x) = sin⁡(x)/x²

Q5. Differentiate the function: m(x) = e^x/(x + 1)

What is the Quotient Rule in Calculus?

Quotient Rule is a formula used to find the derivative of a function that is the ratio of two differentiable functions. It is essential for solving calculus problems involving division of functions.

What is the Quotient Rule Formula?

Quotient Rule formula is written as:

(u/v)’ = (u’v – uv’)/v²

When to Use the Quotient Rule?

We use the Quotient Rule to differentiate a function that is expressed as the division of two differentiable functions. It is particularly useful in complex problems where the numerator and denominator are not easily simplified.

What is the quotient rule with U and V?

Quotient Rule with U and V are: d/dx (u/v) = (v du/dx – u dv/dx)/v