When we differentiate trigonometric functions, the result is another trig function. The first few derivatives in the list below are worth memorizing. Don’t forget that chain rule & friends apply for all functions, not just polynomials. Given the first two rows below, you should be able to derive the rest using just the power rule and the chain rule.
Function
Derivative
sinx
cosx
cosx
−sinx
sin2x
2cos2x
cos2x
−2sin2x
sin2x
sin2x
cos2x
−sin2x
cscx
−cscxcotx
secx
secxtanx
tanx
sec2x
cotx
−csc2x
Example
Let’s differentiate
7cos2(x2−3).
Using the constant rule, we have
dxd7cos2(x2−3)=7⋅dxd(cos(x2−3))2,
and using the power rule and chain rule, we have
7⋅2cos(x2−3)⋅dxdcos(x2−3).
Now we must keep using the chain rule:
7⋅2cos(x2−3)⋅−sin(x2−3)⋅dxdx2−3.
This simplifies to
−7sin(2x2−3)⋅dxdx2−3,
but we aren’t done with the chain rule:
−7sin(2x2−3)⋅2x2−31⋅dxd(x2−3).
There is finally nothing left to differentiate. We have