Question

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

Answer 1

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Answer on Question #64250 – Math – Differential Equations

Question

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

Solution

Recall the following formulae:

$$\begin{array}{l} (uv)' = u'v + uv' \\ \left(f(g(x))\right)' = f'(g(x)) \cdot g'(x) \\ (e^x)' = e^x \\ (\sin(x))' = \cos(x) \\ (\cos(x))' = -\sin(x) \\ (ax)' = a \\ \end{array}$$

Then

$$\begin{array}{l} (e^{ax} \cdot \sin(2x) \cdot \cos(3x))' = ([e^{ax} \cdot \sin(2x)] \cdot \cos(3x))' = (\cos(3x))' \cdot (e^{ax} \cdot \sin(2x)) + (\cos(3x)) \cdot (e^{ax} \cdot \sin(2x))' = \\ = (-3\sin(3x)) \cdot (e^{ax} \cdot \sin(2x)) + (\cos(3x)) \cdot (ae^{ax} \cdot \sin(2x) + 2e^{ax} \cdot \cos(2x)) = \\ = -3\sin(3x) \cdot \sin(2x) \cdot e^{ax} + ae^{ax} \cdot \sin(2x) \cdot \cos(3x) + 2e^{ax} \cdot \cos(2x) \cdot \cos(3x). \end{array}$$

**Answer**: $-3\sin(3x) \cdot \sin(2x) \cdot e^{ax} + ae^{ax} \cdot \sin(2x) \cdot \cos(3x) + 2e^{ax} \cdot \cos(2x) \cdot \cos(3x)$.