Question #a254f

2 Answers
Feb 16, 2018

It should be #d/dx(sin^-1x) = 1/sqrt(1-x^2)#

Explanation:

#y = sin^-1x# so #sin y = x#.

Now use implicit differentiation to get

#dy/dx = 1/cosy#

and use trigonmetry to get

# = 1/sqrt(1-x^2)#

Feb 17, 2018

A more visual yet same as the answer below

Explanation:

#y=sin^-1(x)<=>sin(y)=x#

#sin(y)=x#

Create a triangle where #sin(y)=x/1#

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The missing side can be found using Pythagorean's Theorem

Differentiate both sides implicitly (W.R.T. #x#)

#dy/dx*cos(y)=1#

Divide #cos(y)# from both sides

#dy/dx=1/cos(y)#

Rewrite in terms of #x#

Since #sin(y)=x/1#

Then #cos(y)=sqrt(1-x^2)/1=sqrt(1-x^2)larr# Based on triangle

So #dy/dx=1/cos(y)=1/sqrt(1-x^2)#

#:.dy/dx[arcsin(x)]=1/sqrt(1-x^2)#