## Differentiate Inverse Cos – Proof

**Now available from trevorpythag.blogspot.com**

How to differentiate cos^{-1}x

y=cos^{-1}x

Bring the cos across

cosy = x

Differentiate both sides, remember when differentiating y time by dy/dx

-sin(y) dy/dx = 1

dy/dx = -1/siny

However we want to get the differential in terms of x, to do this we can use the identity

sin^{2}t+cos^{2}t = 1

so

sint = √(1 – cos^{2}t)

putting this into our expression for dy/dx we get

dy/dx = -1/√(1-cos^{2}y)

but cosy = x so

**dy/dx =- 1/√(1-x ^{2})**

## Differentiate Inverse Sine

**Now avalible from trevorpythag.blogspot.com**

This tutorial explain how to differentiate inverse sine, this applies when using radians.

begin with

y = sin^{-1} x

bring sin^{-1} across to become sin

sin y = x

differentiate

cos y dy/dx = 1

note that the derivative of sint wrtt is cos t as explained in an earlier tutorial and by the chain rule when we differentiated sin y it became cosy time dy/dx as we are differnetiatiny a y and the derivative of y is dy/dx

then make dy/dx the subject

dy/dx = 1/cosy

We know the identity

sin^{2}t + cos^{2}t = 1

so we can wrtie

cos t =√(1 – sin^{2}t)

we can now put this into the expression for dy/dx to get

dy/dx = 1/√(1 – sin^{2}y)

but we know from the second line that sin y = x so

** dy/dx = 1/√(1 – x**^{2}**)**

## Comments