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Posts Tagged ‘arcsin’

Differentiate Inverse Cos – Proof

January 8, 2009 7 comments

Now available from trevorpythag.blogspot.com

How to differentiate cos-1x

y=cos-1x
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
sin2t+cos2t = 1
so
sint = √(1 – cos2t)

putting this into our expression for dy/dx we get

dy/dx = -1/√(1-cos2y)
but cosy = x so

dy/dx =- 1/√(1-x2)

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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
sin2t + cos2t = 1
so we can wrtie
cos t =√(1 – sin2t)

we can now put this into the expression for dy/dx to get
dy/dx = 1/√(1 – sin2y)
but we know from the second line that sin y = x so

dy/dx = 1/√(1 – x2)

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