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.

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.

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