prove the derivative of sin is cos
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)
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