Question

If y=e^ax 〖cos〗^(3 ) X 〖sin〗^2 X find dy/dx

Answer 1

Answer on Question #64134 – Math – Differential Equations

Question

If

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

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

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

Answer: $\frac{dy}{dx} = a e^{ax} \cos^3(x) \sin^2(x) - 3 e^{ax} \cos^2(x) \sin^3(x) + 2 e^{ax} \sin(x) \cdot \cos^4(x)$.

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