Question

Y=e^a x.cos^3 x.sin^2 x Find dy/dx

Answer 1

Answer on Question #64191 – Math – Differential Equations

Question

1. $y = e^{ax}\cos^3 x\sin^2 x$. Find $\frac{dy}{dx}$.

Solution

The derivative of the product of three functions is

$$(u \cdot v \cdot w)' = u' \cdot v \cdot w + u \cdot v' \cdot w + u \cdot v \cdot w'$$

If

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

then

$$\begin{aligned} \frac{dy}{dx} = & (e^{ax})' \cos^3(x) \sin^2(x) + e^{ax} \left( \cos^3(x) \right)' \sin^2(x) + e^{ax} \cos^3(x) \left( \sin^2(x) \right)' = \\ & = a e^{ax} \cos^3(x) \sin^2(x) \\ & \quad + e^{ax} \left( 3 \cos^2(x) \cdot (-\sin(x)) \cdot \sin^2(x) + 2 \sin(x) \cos(x) \cdot \cos^3(x) \right) \\ & = \\ & = a e^{ax} \cos^3(x) \sin^2(x) - 3 e^{ax} \cos^2(x) \sin^3(x) + 2 e^{ax} \sin(x) \cdot \cos^4(x) = \\ & = e^{ax} \cos(x) \sin(x) \left( a \cos^2(x) \sin(x) - 3 \cos(x) \sin^2(x) + 2 \cos^3(x) \right) \end{aligned}$$

Answer:

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

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