Differentiating Inverse Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Differentiating inverse trigonometric functions

What are the inverse trigonometric functions?

• arcsin, arccos and arctan are functions defined as the inverse functions of sine, cosine and tangent respectively

  •   which is equivalent to 

  •   which is equivalent to 

What are the derivatives of the inverse trigonometric functions?

• Unlike with other derivatives, these functions and their derivatives look completely unrelated at first

  • Their derivation involves use of the identity 

  • Hence the squares and square roots!

• Note with the derivative of  that  is the same as

How do I show or prove the derivatives of the inverse trigonometric functions?

• For 

  • Rewrite, 

  • Differentiate implicitly, 

  • Rearrange, 

  • Rewrite using the identity , 

  • And , therefore 

• Similarly, for 

  • 

  • 

  • 

  • 

  • 

• For proving the derivative of , see the Worked Example

• Notice how the derivative of is positive but the derivative of  is negative

  • This subtle but crucial difference can be seen in their graphs

    • has a positive gradient for all values of  in its domain

    •  has a negative gradient for all values of in its domain

What do the derivatives of inverse trig functions look like with linear functions of x?

• For linear functions of the form 

• 

  • 

• 

  • 

• 

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• These are not in the formula booklet 

  • they can be derived from chain rule

  • they are not essential to remember

  • they are not commonly used