# What is the derivative of sin 5x?

Jan 19, 2016

$5 \cos 5 x$

#### Explanation:

Use the chain rule.

The chain rule states that, in the case of a sine function,

$\frac{d}{\mathrm{dx}} \left[\sin u\right] = \cos u \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

More generally, the chain rule says to identify an inside function and an outside function. Here, the outside function is $\sin x$, and the inside function is $5 x$.

The chain rule then says to differentiate the outside function, and the derivative of $\sin x$ is $\cos x$. With this derivative, plug in the inside function: this gives us $\cos 5 x$.

The final step of this is to multiply the function by the derivative of the inside function, and the derivative of $5 x$ is $5$.

Thus, the derivative of the whole function is $\cos 5 x \cdot 5$, or $5 \cos 5 x$.

Using the rule given at the top:

$\frac{d}{\mathrm{dx}} \left[\sin 5 x\right] = \cos 5 x \cdot \frac{d}{\mathrm{dx}} \left[5 x\right] = \cos 5 x \cdot 5 = 5 \cos 5 x$