Example 1: Polynomial Function

Given that \(y=x^3\sqrt{5x-2} \,, \quad find \quad \frac{dy}{dx} \)

\begin{align} Let \quad u & = x^3 \quad and \quad v= \sqrt{5x-2}=(5x-2)^{\frac{1}{2}} \\ then \quad \frac{du}{dx} & =3x^2 \\ and \quad \frac{dv}{dx} & =\frac{1}{2}(5)(5x-2)^{-\frac{1}{2}} \\ Using \, the \, & product \, rule, \\ \frac{dy}{dx} & = u\frac{dv}{dx} + v\frac{du}{dx} \\ & = x^3 \left(\frac{5}{2}(5x-2)^{-\frac{1}{2}}\right) + (5x-2)^{\frac{1}{2}} (3x^2) \\ & = \frac{\frac{5}{2}x^3+(5x-2)(3x^2)}{(5x-2)^\frac{1}{2}} \\ \end{align}

Example 2: Trigonometric Function

Given that \(y=e^{3x}cos^2(5x) \,, \quad find \quad \frac{dy}{dx}\)

\begin{align} Let \quad u & = e^{3x} \quad and \quad v= cos^2(5x) \\ then \quad \frac{du}{dx} & = 3e^{3x} \\ and \quad \frac{dv}{dx} & = 2cos(5x) \times 5(-sin(5x)) \\ & = -10cos(5x)sin(5x) \\ Using \, the \, & product \, rule, \\ \frac{dy}{dx} & = u\frac{dv}{dx} + v\frac{du}{dx} \\ & = e^{3x}(-10cos(5x)sin(5x)) + cos^2(5x)3e^{3x}\\ & = e^{3x}(-10cos(5x)sin(5x)+3cos^2(5x)) \\ Using \, the \, & double \, angle \, formula, \\ & = e^{3x}(-5sin(10x)+3cos^2(5x)) \\ \end{align}