HSC Science (Electronics) 12th Board ExamMaharashtra State Board
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# Solution for If y = f(x) is a differentiable function of x such that inverse function x = f^–1 (y) exists, then prove that x is a differentiable function of y and dx/dy=1/(dy/dx) where dy/dx≠0 - HSC Science (Electronics) 12th Board Exam - Mathematics and Statistics

ConceptDerivative Derivative of Inverse Function

#### Question

If y = f(x) is a differentiable function of x such that inverse function x = f–1 (y) exists, then prove that x is a differentiable function of y and dx/dy=1/((dy/dx)) " where " dy/dx≠0

#### Solution 1

Let deltay be the increment in y corresponding to an increment deltax in x.

as deltax->0,deltay->0

Now y is a differentiable function of x.

therefore lim_(deltax->0)(deltay)/(deltax)=dy/dx

Now (deltay)/(deltax)xx(deltax)/(deltay)=1

therefore (deltax)/(deltay)=1/((deltay)/(deltax))

Taking limits on both sides as deltax->0, we get

lim_(deltax->0)(deltax)/(deltay)=lim_(deltax->0)[1/((deltay)/(deltax))]=1/(lim_(dx->0)(deltay)/(deltax))

lim_(deltax->0)(deltax)/(deltay)=1/(lim_(dx->0)(deltay)/(deltax))   ....[as deltax->0,deltay->0]

Since limit in R.H.S. exists

limit in L.H.S. also exists and we have,

lim_(deltay->0)(deltax)/(deltay)=dx/dy

dx/dy=1/(dy/dx), where dy/dxne0

Let y=tan^-1x

x=tany=>cosy=1/sqrt(1+tan^2y)=1/sqrt(1+x^2)

therefore sec^y.dy/dx=1=>dx/dy=sec^2y

dy/dx=1/(dx/dy)=1/sec^2y=cos^2y=>dy/dx=cos^y

(d(tan^-1x))/dx=cos^2y=(cosy)^2=(1/sqrt(1+x^2))^2

therefore d/dx(tan^-1x)=1/(1+x^2)

#### Solution 2

'y’ is a differentiable function of ‘x’.

Let there be a small change δx in the value of ‘x’.

Correspondingly, there should be a small change δy in the value of ‘y’.

As δx → 0, δy → 0

Is there an error in this question or solution?

#### APPEARS IN

2015-2016 (July) (with solutions)
Question 5.1.3 | 2 marks
2016-2017 (March) (with solutions)
Question 5.2.1 | 4 marks

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Solution for question: If y = f(x) is a differentiable function of x such that inverse function x = f^–1 (y) exists, then prove that x is a differentiable function of y and dx/dy=1/(dy/dx) where dy/dx≠0 concept: Derivative - Derivative of Inverse Function. For the courses HSC Science (Electronics), HSC Science (Computer Science), HSC Arts, HSC Science (General)
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