3. Derivatives of the Inverse Trigonometric Functions
by M. Bourne
Recall from when we first met inverse trigonometric functions:
"sin-1x" means "find the angle whose sine equals x".
Example:
If x = sin-10.2588 then by using the calculator, x = 15°. We have found the angle whose sine is 0.2588.
Notation:
We also write: arcsin x to mean the same thing as sin-1x.
It is better to use arcsin x because normally in mathematics, a number raised to the power -1 means the reciprocal. Example: 3-1 = 1/3. Your calculator probably uses the notation: sin-1x. This section mostly uses the sin-1x notation (since it was originally written to be consistent with calculator notation).
You may also wish to go back to background information on inverse trigonmetric equations.
Derivatives of Inverse Trigonometric Functions
The following are the formulas for the derivatives of the inverse trigonometric functions:
Example 1:
Find the derivative of y = cos-15x.
Example 2:
Find the derivative of y = sin-1(1 − x2).
Example 3:
Find 
if x + y = tan-1( x2 + 3y).
Exercises
1. Find the derivative of y = 3 cos-1(x2 + 0.5).
2. Find the derivative of y = 4 tan-13x4.
3. Find the derivative of y = (x2 + 1) sin-14x.
Let's do this problem in LiveMath. You can change the function reasonably safely.
4. Find the derivative of sin-1(x + y) + y = x2.
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