Question

State the chain rule, the product rule and the quotient rule of differentiation. For each rule, give an example of a trigonometric expression which can be differentiated using the rule.

Answer 1

Chain rule of differentiation:

$\dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx}$

Example: $f(x) = sin(3x^2+x)$

It looks like the outside function is the sine and the inside function is 3x²+x. The derivative is then.

$f'(x) = cos(3x^2+x)(6x+1)$

$= (6x+1)cos(3x^2+x)$

Product Rule of Differentiation:

$\dfrac{d(uv)}{dx} = u \dfrac{dv}{dx} + v \dfrac{du}{dx}$

Example : $f(x) = xsinx$

$f'(x) = x\dfrac{d(sinx)}{dx} + sinx \dfrac{d(x)}{dx}$

$f'(x) = xcosx+sinx.$

Quotient Rule of differentiation:

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

Example: $f(x) = \dfrac{sinx}{x}$

$f'(x) = \dfrac{x\dfrac{d(sinx)}{dx}-sinx \dfrac{d(x)}{dx}}{x^2}$

$f'(x) = \dfrac{xcosx-sinx}{x^2}$