2.4. Derivatives of Trigonometric Functions http://www.ck12.org
The derivatives of the remaining trigonometric functions are shown in the table below.
Derivatives of Trigonometric Functions
d
dx[sinx] =cosx
d
dx[cosx] =−sinx
d
dx[tanx] =sec(^2) x
d
dx[secx] =secxtanx
d
dx[cscx] =−cscxcotx
d
dx[cotx] =−csc
(^2) x
Keep in mind that for all the derivative formulas for the trigonometric functions, the argumentxis measured in
radians.
Example 1:
Show thatdxd[tanx] =sec^2 x.
Solution:
It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.
Since
tanx=sincosxx,
then
d
dx[tanx] =
d
dx
[sinx
cosx
]
Using the quotient rule,
=(cosx)(cosxcos)− 2 (xsinx)(−sinx)
=cos(^2) x+sin (^2) x
cos^2 x
=cos^12 x
=sec^2 x
Example 2:
Findf′(x)iff(x) =x^2 cosx+sinx.
Solution:
Using the product rule and the formulas above, we obtain