What is the derivative of ##sin 5x##?

##5cos5x##

Use the .

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##