Implicit differentiation

Question

Answer 1

a)

Given equation:

(1/x)+(1/y)=3

Take derivative both sides with respect to x

d/dx(1/x)+d/dx(1/y)=d/dx(3)

d/dx(x^-1)+d/dx(y^-1)=d/dx(3)

Use formulas: d/dx(x^n)=nx^(n-1)  and d/dx(c)=0

(-1)x^(-1-1)+(-1)y^(-1-1)dy/dx=0

-x^(-2)-y^(-2)dy/dx=0

dy/dx =-x^(-2)/y^(-2)

         =-y^2/x^2

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b)

3cosxsiny=1

Take derivative both sides with respect to x

d/dx(3cosxsiny)=d/dx(1)

3d/dx(cosxsiny)=d/dx(1)

Use product rule:d/dx(uv)=uv'+vu'

u=cosx    u' =-sinx

v = siny     v'= cosy dy/dx

3(cosx*cosy dy/dx+siny(-sinx))=0

cosx*cosy dy/dx-sinxsiny=0

dy/dx=sinxsiny/cosxcosy

          =tanxtany 

 

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