Quotient Rule Examples
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 = (ex + 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 . {ex + (1/x)} – (ex + log x)(3cos3x)]/sin23x
= [{ex + (1/x)}.sin3x – 3(ex + log x)cos3x ]/sin23x
Example 6: Find the derivative of y = (ex + e-x)/ (ex – 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]
= [(ex – e-x)(ex – e-x) – (ex + e-x)(ex + e-x)]/(ex – e-x)2
= [(ex – e-x)2 – (ex + e-x)2]/(ex – e-x)2
= 4/(ex – e-x)2
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.
Table of Content
- What is Quotient Rule?
- How to Use Quotient Rule?
- Quotient Rule Examples
- Practice Questions on Quotient Rule
- FAQs on Quotient Rule