1) if x=a(1+cos θ),y=a( θ+sin θ),find (d^2)y/d(x^2) at  θ= π/2 π=pi 2) if x=a(cos θ+ θsin θ),y=a(sin θ- θcos θ),find (d^2)y/d(x^2) 3)y=x+tanx,prove (cos^2)x *...

1) if x=a(1+cos θ),y=a( θ+sin θ),find (d^2)y/d(x^2) at  θ= π/2

π=pi

2) if x=a(cos θ+ θsin θ),y=a(sin θ- θcos θ),find (d^2)y/d(x^2)

3)y=x+tanx,prove (cos^2)x * (d^2)y/d(x^2)-2y +2x =0

4)if ylogx=x-y,prove dy/dx=logx/(1+logx)^2

5)if sec(x+y/x-y)=a,prove dy/dx=y/x

6)if y=xsiny,prove dy/dx =y/x(1-xcosy)

7)find dy/dx when

i)x^2/a^2+y^2/b^2=1

ii)sin(x+y)=2/3

iii)xy^3-x^3y=x

iv)(x^2+y^2)^2=xy

v)ysecx+tanx+x^2y=0

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embizze | High School Teacher | (Level 1) Educator Emeritus

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3) y=x+tanx : prove that cos^2x(y'')-2y+2x=0

y'=1+sec^2x and y''=2secx(secxtanx)=2sec^2xtanx

Then cos^2x (y'')-2y+2x

=cos^2x (2sec^2xtanx)-2y+2x

=2tanx-2y+2x    

=2(x+tanx)-2y    but x+tanx=y so

=2y-2y

=0 as required.

6) y=xsiny  : show that y'=y/(x(1-xcosy))

y=xsiny

y'=siny+xcosyy'

y'-xcosyy'=siny

y'(1-xcosy)=siny

y'=(siny)/(1-xcosy)      but siny=y/x so

y'=y/(x(1-xcosy)) as required.

7i) x^2/a^2+y^2/b^2=1; find y'

b^2x^2+a^2y^2=a^2b^2

2b^2x+2a^2yy'=0

2a^2yy'=-2b^2x

y'=(-b^2x)/a^2y)

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