Answer to Question #188363 in Calculus for Maureen

Question

use the technique of derivative to find dy/dx

y=3 sin^5 x

Answer 1

In this case, we use the Chain Rule.

$$y=3\sin^5 x = 3(\sin x)^5$$

Let $u=\sin x$ so that $y=3u^5$

Taking derivative of $u$ with respect to $x$ , we have:

$\frac{du}{dx} = \cos x$

Similarly, the derivative of $y$ with respect to $u$ is given as

$\frac{dy}{du} = 15u^4$

By Chain Rule,

$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

Therefore, $\frac{dy}{dx} = 15(\sin x)^4 \cos x$