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If y=x^3logx ,"p r o v et h a t"(d^4y)/(...

If `y=x^3logx ,"p r o v et h a t"(d^4y)/(dx^4)=6/xdot`

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`y=x^{3} log x`

`frac{d y}{d x}=x^{2}+3 x^{2} log x`

`frac{d^{2} y}{d x^{2}}=2 x+6 x log x+3 x=5 x+6 x log x`

`frac{d^{3} y}{d x^{3}}=5+6(log x+1)=11+6 log x`

`frac{d^{4} y}{d x^{4}}=frac{6}{x}`

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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=e^-x cos x, show that (d^2y)/(dx^2)= 2e^-1 sin x

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  2. If y=log(sinx),"p r o v et h a t"(d^3y)/(dx^3)=2cosxcos e c^3xdot

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  3. If y=x^3logx ,"p r o v et h a t"(d^4y)/(dx^4)=6/xdot

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  4. If y=(logx)/x ,s howt h a t(d^2y)/(dx^2)=(2logx-3)/(x^3)

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  5. If y=2 sin x+3cos x, show that (d^2y)/(dx^2)+y=0

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  6. Find . (d^2y)/(dx^2),ifx=a t^2,y=2a tdot

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  7. If x=sin theta,y=cosptheta,p rov et h a t (1-x^2)y2-x y1+p^2y=0,w ...

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  8. If y=tan^-1x find (d^2y)/(dx^2) in terms of y alone.

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  9. If x^m y^n=(x+y)^(m+n), prove that (d^2y)/(dx^2)=0

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  10. If y^3-y=2x ,p rov et h a t(d^2y)/(dx^2)=(-24 y)/((3y^2-1)^3)

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  11. If y=x^x ,fin d(d^2y)/(dx^2)

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  12. If y=sin^(-1)x ,s howt h a t(d^2y)/(dx^2)=x/((1-x^2)^(3/2))

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  13. If y=Acosn x+Bsinn x ,s howt h a t (d^2y)/(dx^2)+n^2y=0

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  14. If y=tanx+secx ,p rov et h a t(d^2y)/(dx^2)=(cosx)/((1-sinx)^2)

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  15. If y=(a x+b)/(x^2+c), then (2x y1+y) y3=

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  16. If y=(log)e(x/a)^x ,then x^3y2=

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  17. If sin^(-1) (1 – x) – 2sin^(-1) x = π / 2,then x is equal to

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  18. If y=2/(sqrt(a^2-b^2)){tan^(-1)((a-b)/(a+b))tanx/2} , then show that (...

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  19. If y=x^(n-1)lnx then x^2y2+(3-2n)x y1 is equal to -(n-1)^2y (b) (n...

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  20. If x y-(log)e y=1 satisfies the equation x(y y2+y1 2)-y2+lambday y1=0,...

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