Find the value

Question:

Differentiate w.r.t $x: e^{2 x} \sin 3 x$

 

Solution:

Let $y=e^{2 x} \sin 3 x, z=e^{2 x}$ and $w=\sin 3 x$

Formula :

$\frac{d\left(e^{x}\right)}{d x}=e^{x}$ and $\frac{d(\sin x)}{d x}=\cos x$

According to product rule of differentiation

$\mathrm{dy} / \mathrm{dx}=\mathrm{w} \times \frac{\mathrm{dz}}{\mathrm{dx}}+\mathrm{z} \times \frac{\mathrm{dw}}{\mathrm{dx}}$

$=\left[\sin 3 x \times\left(2 \times e^{2 x}\right)\right]+\left[e^{2 x} \times 3 \cos 3 x\right]$

$=e^{2 x} \times[2 \sin 3 x+3 \cos 3 x]$

 

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