Differentiation of Trigonometric Functions

Example 1:

$$ y=\sin 4x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=\sin 4x$$ $$ \frac{dy}{dx}=\frac{d}{dx}\sin 4x$$ $$ \textup{Let}\; u= 4x$$ $$ \frac{dy}{dx}=\frac{d}{dx} \sin u$$ $$ \frac{dy}{dx}=\frac{d}{du} \sin u \frac{du}{dx}$$ $$ \because\frac{d}{dx}\sin x=\cos x$$ $$ \therefore\frac{dy}{dx}=\cos u \frac{d}{dx}4x$$ $$ \frac{dy}{dx}=\cos 4x (4\frac{d}{dx}x)$$ $$ \because\frac{d}{dx} x=1 $$ $$ \therefore \frac{dy}{dx}=\cos 4x (4(1))$$ $$ \frac{dy}{dx}=\cos 4x (4)$$ $$ \frac{dy}{dx}=4\cos 4x $$