Example 1: Polynomial Function

Given that $y = x^{3} \sqrt{5 x - 2}, f i n d \frac{d y}{d x}$

\begin{aligned} L e t u & = x^{3} a n d v = \sqrt{5 x - 2} = (5 x - 2)^{\frac{1}{2}} \\ t h e n \frac{d u}{d x} & = 3 x^{2} \\ a n d \frac{d v}{d x} & = \frac{1}{2} (5) (5 x - 2)^{- \frac{1}{2}} \\ U s i n g t h e & p r o d u c t r u l e, \\ \frac{d y}{d x} & = u \frac{d v}{d x} + v \frac{d u}{d x} \\ = x^{3} (\frac{5}{2} (5 x - 2)^{- \frac{1}{2}}) + (5 x - 2)^{\frac{1}{2}} (3 x^{2}) \\ = \frac{\frac{5}{2} x^{3} + (5 x - 2) (3 x^{2})}{(5 x - 2)^{\frac{1}{2}}} \end{aligned}

Example 2: Trigonometric Function

Given that $y = e^{3 x} c o s^{2} (5 x), f i n d \frac{d y}{d x}$

\begin{aligned} L e t u & = e^{3 x} a n d v = c o s^{2} (5 x) \\ t h e n \frac{d u}{d x} & = 3 e^{3 x} \\ a n d \frac{d v}{d x} & = 2 c o s (5 x) \times 5 (- s i n (5 x)) \\ = - 10 c o s (5 x) s i n (5 x) \\ U s i n g t h e & p r o d u c t r u l e, \\ \frac{d y}{d x} & = u \frac{d v}{d x} + v \frac{d u}{d x} \\ = e^{3 x} (- 10 c o s (5 x) s i n (5 x)) + c o s^{2} (5 x) 3 e^{3 x} \\ = e^{3 x} (- 10 c o s (5 x) s i n (5 x) + 3 c o s^{2} (5 x)) \\ U s i n g t h e & d o u b l e a n g l e f o r m u l a, \\ = e^{3 x} (- 5 s i n (10 x) + 3 c o s^{2} (5 x)) \end{aligned}