What is the derivative of #sin(3x)#?
The chain rule is a tool for differentiating composite functions, that is, a function inside a function.
Here, we have
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, the first part of the chain rule, the differentiated outside function with the inside function unchanged, gives us
We can generalize this to all derivatives of sine functions: