# 3. Derivatives of the Inverse Trigonometric Functions

by M. Bourne

Recall from when we first met inverse trigonometric functions:

"sin^{-1}x" means "find the angle whose sine equalsx".

### Example 1

If *x* =
sin^{-1}0.2588 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

^{-1}

*x*.

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^{-1}*x* 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^{-1}5*x*.

### Example 3

Find the derivative
of *y* = sin^{-1}(1 − *x*^{2}).

### Example 4

Find `(dy)/(dx)` if *x *+ *y* = tan^{-1}( *x*^{2} + 3*y*).

## Exercises

1. Find the derivative of *y* = 3 cos^{-1}(*x*^{2} + 0.5).

2. Find the derivative of *y* = 4 tan^{-1}3*x*^{4}.

3. Find the derivative of *y* = (*x*^{2} + 1) sin^{-1}4*x*.

4. Find the derivative of sin^{-1}(*x *+ *y*) + *y *= *x*^{2}.

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