# What is the derivative of sin(3x)?

Jun 14, 2016

$3 \cos \left(3 x\right)$

#### Explanation:

The chain rule is a tool for differentiating composite functions, that is, a function inside a function.

Here, we have $\sin \left(3 x\right)$. This can be thought of as the function $3 x$ being put inside of the function $\sin \left(x\right)$.

When finding the derivative of such a function, the chain rule tells us that the derivative will be equal to the derivative of the outside function with the original inside function still inside of it, all multiplied by the derivative of the inside function.

So, for $\sin \left(3 x\right)$, the derivative the $\sin \left(x\right)$, the outside function, is $\cos \left(x\right)$.

So, the first part of the chain rule, the differentiated outside function with the inside function unchanged, gives us $\cos \left(3 x\right)$. Then, this is multiplied by the derivative of the inside function. The derivative of $3 x$ is $3$, so the overall derivative is $\cos \left(3 x\right) \times 3 = 3 \cos \left(3 x\right)$.

We can generalize this to all derivatives of sine functions:

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