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If y=x^3logx , prove that (d^4y)/(dx^4)=...

If `y=x^3logx` , prove that `(d^4y)/(dx^4)=6/x` .

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To prove that \(\frac{d^4y}{dx^4} = \frac{6}{x}\) for the function \(y = x^3 \log x\), we will differentiate \(y\) four times step by step. ### Step 1: First Derivative Given: \[ y = x^3 \log x \] Using the product rule: ...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=e^(-x)cosx, show that (d^(2)y)/(dx^(2))=2e^(-x)sinx.

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

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  3. If y=x^3logx , prove that (d^4y)/(dx^4)=6/x .

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  4. If y=log(sinx) , prove that (d^3y)/(dx^3)=2cosx cos e c^3x .

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

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

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  7. If x=asectheta , y=btantheta , prove that (d^2y)/(dx^2)=-(b^4)/(a^2\ y...

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  8. If x=a(costheta+thetasintheta) , y=a(sintheta-thetacostheta) , prove t...

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  9. If y=e^xcosx , prove that (d^2y)/(dx^2)=2\ e^xcos(x+pi/2) .

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  10. If x=acostheta , y=bsintheta , show that (d^2y)/(dx^2)=-(b^4)/(a^2y^3)...

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  11. If x=a(1-cos^3theta) , y=a\ s in^3theta , prove that (d^2y)/(dx^2)=(32...

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

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

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

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

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  16. If x=costheta , y=sin^3theta , prove that y\ (d^2y)/(dx^2)+((dy)/(dx))...

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  17. If y=sin(sinx) , prove that (d^2y)/(dx^2)+tanxdot(dy)/(dx)+y\ cos^2x=0...

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  18. If x=sint ,y=sinp t ,"p r o v et h a t" (1-x^2)(d^2y)/(dx^2)-x(dy)/...

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  19. If y=(sin^(-1)x)^2 , prove that (1-x^2)y2-x y1-2=0 .

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  20. If y=e^tan^((-1)x) , prove that (1+x^2)y2+(2x-1)y1=0 .

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