Question

What is the derivative of `sin 5x`?

Answer 1

`5cos5x`

Explanation:

Use the chain rule.

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

`d/dx[sinu]=cosu*(du)/dx`

More generally, the chain rule says to identify an inside function and an outside function. Here, the outside function is `sinx`, and the inside function is `5x`.

The chain rule then says to differentiate the outside function, and the derivative of `sinx` is `cosx`. With this derivative, plug in the inside function: this gives us `cos5x`.

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

Thus, the derivative of the whole function is `cos5x*5`, or `5cos5x`.

Using the rule given at the top:

`d/dx[sin5x]=cos5x*d/dx[5x]=cos5x*5=5cos5x`