What is the derivative of x sinx

Derivative Proof of sin(x)

We can prove the derivative of sin(x) using the limit definition and the double angle formula for trigonometric functions.

Derivative proof of sin(x)

For this proof, we can use the limit definition of the derivative.

Limit Definition for sin:

Using angle sum identity, we get

Rearrange the limit so that the sin(x)'s are next to each other

Factor out a sin from the quantity on the right

Seperate the two quantities and put the functions with x in front of the limit (We are only concerned with the limit of h)

We can see that the first limit converges to 1

and the second limit converges to 0.

We can plug in 1 and 0 for the limits and get cos(x)