Differentiation Techniques – Derivatives of Inverse Trigonometric Functions

There is a total of six trigonometric functions:

  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Cosecant (csc)
  • Secant (sec)
  • Cotangent (cot)

Each of the six trigonometric functions have their inverse functions and are written as the following:

  • Sin-1(u)
  • Cos-1(u)
  • Tan-1(u)
  • Csc-1(u)
  • Sec-1(u)
  • Cot-1(u)

u is the expression inside the trigonometric function.

All the inverse trigonometric functions have their own derivatives. These formulas already have the chain rule built inside of them.

To obtain the derivative of each inverse trigonometric function, simply find your u expression, take the derivative of u, and plug in u and u’ inside the derivative formula.

Example 1:

Find F’(x) if F(x) = cos-1(3x)

Solution:

Use the formula of the derivative of cos-1 (u):

Following the formula,

u = 3x

u’ = 3

Example 2:

Find the Derivative of the function F(y) = tan-1( \sqrt{y} ).

Solution:

We can rewrite F(y) = tan-1( \sqrt{y} ) as tan-1(y1/2). This will help us to find u’.

Use the formula of the derivative of tan-1 (u):

u = y1/2

u’ = 1/2y-1/2

Example 3:

Find the derivative of the function F(y) = 2y + sin-1(y)

Solution:

We will take the derivative of each term:

First, let’s take the derivative of 2y, which is simply 2.

Second, let’s take the derivative of sin-1(y).

We will use the formula for the derivative of inverse sine:

u = y

u’ = y’ = 1

Substituting u and u’ into the formula,

References:
https://www.cliffsnotes.com/study-guides/calculus/calculus/the-derivative/differentiation-of-inverse-trigonometric-functions
https://tutorial.math.lamar.edu/classes/calci/DiffInvTrigFcns.aspx
https://www.math24.net/derivatives-inverse-trigonometric-functions/
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/invtrigderivdirectory/InvTrigDeriv.html

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