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 1

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`.

Most calculators use the (confusing) notation: `sin^-1 x`. This section mostly uses the sin-1x notation (since it was originally written to be consistent with calculator notation), however you are encouraged to use the superior notation, `arcsin\ x`.

You may also wish to go back to background information on inverse trigonmetric equations.

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Derivatives of Inverse Trigonometric Functions

The following are the formulas for the derivatives of the inverse trigonometric functions:

`(d(sin^-1u))/(dx)=1/sqrt(1-u^2)(du)/(dx)`

`(d(cos^-1u))/(dx)=(-1)/sqrt(1-u^2)(du)/(dx)`

`(d(tan^-1u))/(dx)=1/(1+u^2)(du)/(dx)`

Example 2

Find the derivative of y = cos-15x.

Example 3

Find the derivative of y = sin-1(1 − x2).

Example 4

Find `(dy)/(dx)` 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.

4. Find the derivative of sin-1(x + y) + y = x2.