Quotient Rule

The quotient rule is a differentiation technique used to find the derivative of a quotient of two functions. It's particularly useful when dealing with functions that are divided by one another, and it complements the product rule and chain rule in calculus.

If a function can be written as :

y=\frac{u}{v}

then the first derivative can be written as :

\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}

Example

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

First, identify the denominator and numerator :

u=x^2+1

v=x^2-1

Differentiate each function :

u'=2x

v'=2x

Apply quotient rule :

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

Simplify :

\frac{dy}{dx} = \frac{2x(x^2 - 1 - x^2 - 1)}{(x^2 - 1)^2} = \frac{2x(-2)}{(x^2 - 1)^2} = \frac{-4x}{(x^2 - 1)^2}

Example

infoNote

Differentiate $y= \frac{e^x}{\ln(x)}$

Identify the denominator and numerator :

u=e^x

v=\ln{(x)}

Differentiate each function :

u'=e^x

v'=\frac{1}{x}

Apply quotient rule :

\frac{dy}{dx} = \frac{\ln(x) \cdot e^x - e^x \cdot \frac{1}{x}}{[\ln(x)]^2}

Simplify :

\frac{dy}{dx} = \frac{e^x \left(\ln(x) - \frac{1}{x}\right)}{[\ln(x)]^2}

Example

infoNote

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

Identify the denominator and numerator :

u=x^2

v=e^{3x}

Differentiate each function :

u'=2x

v'=3e^{3x}

Apply quotient rule :

\frac{dy}{dx} = \frac{e^{3x} \cdot 2x - x^2 \cdot 3e^{3x}}{(e^{3x})^2}

Simplify :

\frac{dy}{dx} = \frac{e^{3x}(2x - 3x^2)}{e^{6x}} = \frac{2x - 3x^2}{e^{3x}}

Quotient Rule Simplified Revision Notes for Leaving Cert Mathematics

Quotient Rule

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

Differentiate $y= \frac{e^x}{\ln(x)}$

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

500K+ Students Use These Powerful Tools to Master Quotient Rule For their Leaving Cert Exams.

Other Revision Notes related to Quotient Rule you should explore

Quotient Rule Simplified Revision Notes for Leaving Cert Mathematics

Quotient Rule

Differentiate y=x2+1x2−1y=\frac{x^2+1}{x^2-1}y=x2−1x2+1​

Differentiate y=exln⁡(x)y= \frac{e^x}{\ln(x)}y=ln(x)ex​

Differentiate y=x2e3xy= \frac{x^2}{e^{3x}}y=e3xx2​

500K+ Students Use These Powerful Tools to Master Quotient Rule For their Leaving Cert Exams.

Other Revision Notes related to Quotient Rule you should explore

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

Differentiate $y= \frac{e^x}{\ln(x)}$

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