Question

If y = e^ax cos^3 x sin^2 x find dy/dx

Answer 1

If $y = e^{ax} \cos^3(x) \sin^2(x)$ find $\frac{dy}{dx}$.

Solution:

We'll use next rules of differentiation

1) If $y = f(x) \cdot g(x) \cdot r(x)$ then

$$\frac{dy}{dx} = \frac{df(x)}{dx} \cdot g(x) \cdot r(x) + f(x) \cdot \frac{dg(x)}{dx} \cdot r(x) + f(x) \cdot g(x) \cdot \frac{dr(x)}{dx}$$

2) If $y = F(p(x))$ then

$$\frac{dy}{dx} = \frac{dF(p)}{dp} \cdot \frac{dp(x)}{dx}$$

Thus we have

$$\begin{aligned} \frac{dy}{dx} &= \frac{de^{ax}}{dx} \cos^3(x) \sin^2(x) + e^{ax} \frac{d(\cos^3(x))}{dx} \sin^2(x) + e^{ax} \cos^3(x) \frac{d(\sin^2(x))}{dx} = \\ &= a e^{ax} \cos^3(x) \sin^2(x) - 3 e^{ax} \cos^2(x) \sin(x) \sin^2(x) + \\ &\quad + 2 e^{ax} \cos^3(x) \sin(x) \cos(x) = \\ &= e^{ax} \cos^2(x) \sin(x) \left( a \cdot \cos(x) \sin(x) - 3 \sin^2(x) + 2 \cos^2(x) \right) = \\ &= e^{ax} \cos^2(x) \sin(x) \left( \frac{a}{2} \sin(2x) - 3(1 - \cos^2(x)) + 2 \cos^2(x) \right) = \\ &= e^{ax} \cos^2(x) \sin(x) \left( -3 + \frac{a}{2} \sin(2x) + 5 \cos^2(x) \right) \end{aligned}$$

Answer:

$$\frac{dy}{dx} = e^{ax} \cos^2(x) \sin(x) \left( -3 + \frac{a}{2} \sin(2x) + 5 \cos^2(x) \right)$$