# What is the derivative of sin 5x?

Apr 2, 2018

$\frac{d}{\mathrm{dx}} \sin \left(5 x\right) = 5 \cos \left(5 x\right)$

#### Explanation:

When applying the Chain Rule to trigonometric functions such as sine,

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

In this case, $u = 5 x ,$ and so

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

$\frac{d}{\mathrm{dx}} \sin \left(5 x\right) = 5 \cos \left(5 x\right)$

Apr 2, 2018

$5 \cos \left(5 x\right)$

#### Explanation:

We're dealing with a composite function, and whenever we want to differentiate composite functions, we use the Chain Rule stated below:

$f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Our composite function is $\sin \left(5 x\right)$, where:

$f \left(x\right) = \sin x$ and $g \left(x\right) = 5 x$

$f ' \left(x\right) = \cos x$ and $g ' \left(x\right) = 5$

Now we just plug in! We get:

$\cos \left(5 x\right) \cdot 5$

Which we can rewrite as

$5 \cos \left(5 x\right)$