Example 1: Polynomial Function

Given that \(y=\frac{2x}{3x-4} \,, \quad find \quad \frac{dy}{dx} \)

\begin{align} Let \quad u & = 2x \quad and \quad v= 3x-4 \\ then \quad \frac{du}{dx} & =2 \\ and \quad \frac{dv}{dx} & =3 \\ Using \, the \, & quotient \, rule, \\ \frac{dy}{dx} & = \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} \\ & = \frac{(3x-4)(2)-2x(3)}{(3x-4)^2}\\ & = \frac{6x-8-6x}{(3x-4)^2} \\ & = \frac{-8}{(3x-4)^2} \end{align}

Example 2: Trigonometric Function

Given that \(y=\frac{cos(x)}{e^4x} \,, \quad find \quad \frac{dy}{dx}\)

\begin{align} Let \quad u & = cos(x) \quad and \quad v= e^{4x} \\ then \quad \frac{du}{dx} & = -sin(x) \\ and \quad \frac{dv}{dx} & = 4e^4x \\ Using \, the \, & quotient \, rule, \\ \frac{dy}{dx} & = \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} \\ & = \frac{e^{4x}(-sin(x))-cos(x)4e^{4x}}{(e^{4x})^2}\\ & = \frac{e^{4x}(-sin(x)-4cos(x))}{e^{8x}}\\ & =- \frac{(sin(x)+4cos(x))}{e^{4x}}\\ \end{align}