Differentiate the following with respect to x:

Question:
Differentiate the following with respect to x:

$\cos 3 x \sin 5 x$

Solution:

To Find: Differentiation

NOTE : When 2 functions are in the product then we used product rule i.e

$\frac{\mathrm{d}(\mathrm{u} \cdot \mathrm{v})}{\mathrm{dx}}=\mathrm{V} \frac{\mathrm{du}}{\mathrm{dx}}+\mathrm{u} \frac{\mathrm{dv}}{\mathrm{dx}}$

Let us take y = cos 3x sin 5x

So, by using the above formula, we have

$\frac{d}{d x}(\cos 3 x \sin 5 x)=\sin 5 x \frac{d(\cos 3 x)}{d x}+\cos 3 x \frac{d(\sin 5 x)}{d x}=$

$\sin 5 x(-3 \sin 3 x)+\cos 3 x(5 \cos 5 x)=5 \cos (3 x) \cos (5 x)-3 \sin (5 x) 3 \sin (3 x)$

Differentiation of $y=\cos 3 x \sin 5 x$ is $5 \cos (3 x) \cos (5 x)-3 \sin (5 x) 3 \sin (3 x)$