Product Rule (Leaving Cert Mathematics): Revision Notes

Product Rule

The product rule is a fundamental differentiation technique used when you need to differentiate the product of two functions. It allows you to find the derivative of a product without having to multiply the functions first, which can simplify the differentiation process, especially when dealing with complex functions.

If a function can be written as :

$$y=uv$$

Then the first derivative is given by :

$$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$$

Example

Example

Example

$v'$ (using chain rule)

Apply product rule :

$$\frac{dy}{dx} = u'v + uv'$$ $$= (3x+1)^{\frac{1}{2}} (2x) + x^2 \left( \frac{3}{2} (3x+1)^{-\frac{1}{2}} \right)$$ $$= 2x (3x+1)^{\frac{1}{2}} + \frac{3x^2}{2} (3x+1)^{-\frac{1}{2}}$$ $$= x (3x+1)^{-\frac{1}{2}} \left[ 2(3x+1) + \frac{3x}{2} \right]$$ $$= x (3x+1)^{-\frac{1}{2}} \left[ 6x + 2 + \frac{3x}{2} \right]$$ $$= x (3x+1)^{-\frac{1}{2}} \left[ \frac{15x+2}{2} \right]$$ $$= \frac {1}{2}x (3x+1)^{-\frac{1}{2}} \left( 15x+4\right)$$ $$= \frac {15x^2+4x}{2 \sqrt{3x+1}}$$

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