By applying the differentiation rules we have learned so far, we can find the derivatives of trigonometric functions. The differentiation of the six basic trigonometric functions (which are sin,cos,tan,csc,sec, and cot) can be done as shown below:
(1) For y=sinx, we use sina−sinb=2cos(2a+b)sin(2a−b) to get y′:
y′=h→0limhsin(x+h)−sinx=h→0limh2cos(x+2h)sin2h=h→0limcos(x+2h)⋅2hsin2h=cosx.(since x→0limxsinx=1)
(2) For y=cosx, we use cosa−cosb=−2sin(2a+b)sin(2a−b) to get y′:
y′=h→0limhcos(x+h)−cosx=h→0limh−2sin(x+2h)sin2h=−h→0limsin(x+2h)⋅2hsin2h=−sinx.(since x→0limxsinx=1)
(3) For y=tanx, we convert it to cosxsinx and use the quotient rule, which gives
y′=(tanx)′=(cosxsinx)′=cos2xcosx⋅cosx−sinx⋅(−sinx)=cos2x1=sec2x.(since sin2x+cos2x=1)
(4) For y=cotx, we use the same method as for y=tanx, which gives
y′=(cotx)′=(sinxcosx)′=sin2x−sinx⋅sinx−cosx⋅cosx=sin2x−1=−csc2x.
(5) For y=secx, we have
y′=(secx)′=(cosx1)′=cos2x0−1⋅(−sinx)=cos2xsinx=secx⋅tanx.
(6) For y=cscx, we have
y′=(cscx)′=(sinx1)′=sin2x0−1⋅(cosx)=sin2x−cosx=−cscx⋅cotx.
The box below summarizes the derivative of the six trigonometric functions, which of course should be memorized:
- y=sinx⟹y′=cosx
- y=cosx⟹y′=−sinx
- y=tanx⟹y′=sec2x
- y=cotx⟹y′=−csc2x
- y=secx⟹y′=secx⋅tanx
- y=cscx⟹y′=−cscx⋅cotx