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"If "x=(2costheta-cos 2theta)and y=(2sin...

`"If "x=(2costheta-cos 2theta)and y=(2sin theta-sin 2theta)," find "((d^(2))/(dx^(2)))_(theta=(pi)/(2))`

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Verified by Experts

The correct Answer is:
`(-3)/(2)`
`(dy)/(dx)=((dy//d theta))/((dx//d theta))=tan.(3 theta)/(2)`
`rArr(d^(2)y)/(dx^(2))=(3)/(2)sec^(2)(3theta)/(2).(d theta)/(dx)=(3)/(2)sec^(2).(3theta)/(2).(1)/(2(sin 2theta-sin theta))`
`rArr((d^(2)y)/(dx^(2)))_(theta=(pi)/(2))=(3)/(4)sec^(2).(3pi)/(4).(1)/((sin pi-sin.(pi)/(2)))=(3)/(4)xx2xx(1)/((0-1))=(-3)/(2).`
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RS AGGARWAL-DIFFERENTIATION-Exercise 10I
  1. If x=2costheta-cos2\ theta and y=2sintheta-sin2\ theta , prove that (d...

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  2. If x=sin^3t/(sqrtcos2t), y=cos^3t/sqrt(cos2t) show that dy/dx =0 at t=...

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  3. "If "x=(2costheta-cos 2theta)and y=(2sin theta-sin 2theta)," find "((d...

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  4. If x=a(theta-sintheta) , y=a(1+costheta) find (d^2y)/(dx^2)

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  5. Find the second derivative of : (i)x^(11)" "(ii)5^(x)" "(ii...

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  6. Find the second derivative of: (i)x sin x" "(ii)e^(2x)cos 3x" ...

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  7. If y=x+tanx , show that cos^2x\ (d^2y)/(dx^2)-2y+2x=0 .

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  8. If y=2sinx+3cosx ,show that (d^2y)/(dx^2)+y=0

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  9. If y=a cos (logx)+b sin (logx), prove that x^(2)y(2)+xy(1)+y=0.

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  10. If y=e^(-x)cosx, show that (d^(2)y)/(dx^(2))=2e^(-x)sinx.

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  11. If y=sec x-tanx,show that (cosx)(d^(2)y)/(dx^(2))=y^(2).

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  12. If y=("cosec "x+cotx), prove that (sinx)(d^(2)y)/(dx^(2))-y^(2)=0.

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  13. If y=tan^(-1)x , show that (1+x^2)(d^2y)/(dx^2)+2x(dy)/(dx)=0 .

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  14. If y=sin(sinx) then prove that (d^2y)/(dx^2)+tanx. (dy)/(dx)+y cos^2...

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  15. If y=a cos (logx)+b sin (logx), prove that x^(2)y(2)+xy(1)+y=0.

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  16. Find the second derivative of e^(3x)sin4x.

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  17. Find the second derivative of sin 3x cos 5x.

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  18. If y=e^tanx then prove that: cos^2x(d^2y)/(dx^2)-(1+sin2x)(dy)/dx=0

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  19. If y=(logx)/x , show that (d^2y)/(dx^2)=(2logx-3)/(x^3) .

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  20. If y=e^(ax)cosb x, Show that (d^2y)/(dx^2)-2a(dy)/(dx)+(a^2+b^2)y=0

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