Question

if y=e^ax cos^3 x sin^2 x.find dy/dxif y=e^ax cos^3 x sin^2 x.find dy/dx

Answer 1

Find

$$\frac{dy}{dx}$$

if

$$y(x) = e^{ax} \cos^3(x) \sin^2(x).$$

Solution:

We have

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

We'll use that

$$\frac{1}{2} \sin(2x) = \sin(x) \cos(x);$$ $$\sin^2(x) = 1 - \cos^2(x).$$

So

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

Answer:

$$\frac{dy}{dx} = \frac{1}{2} e^{ax} \sin(2x) \cos(x) \left( \frac{a}{2} \sin(2x) + 5 \cos^2(x) - 3 \right)$$