Question

If y = eax cos3x sin2x
find dy/dx.

Answer 1

Answer on Question #62322 – Math – Calculus

Question

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

find $\frac{dy}{dx}$.

Solution

We shall use the following differentiation rules:

1) The product rule:

$$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x);$$

2) The chain rule:

$$(f(g(x))' = f'(g(x)) \cdot g'(x)$$

3) The constant factor rule:

$$(af(x))' = a \cdot f'(x).$$

We shall use the well-known table derivatives:

$$(e^x)' = e^x, \quad (\cos x)' = -\sin x, \quad (\sin x)' = \cos x, \quad x' = 1.$$

Then

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

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

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