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If y=x^x , prove that (d^2y)/(dx^2)-1/y(...

If `y=x^x ,` prove that `(d^2y)/(dx^2)-1/y((dy)/(dx))^2-y/x=0`

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We have given
`y=x^x`
On taking logarithm both sides, we get
`lny=ln(x^x).............(1)`
`lny=xlnx`
On differentiating w.r.t x, we get
`(1/y)(dy)/(dx)=xxx1/x+lnxxx1`
`(1/y)(dy)/(dx)=1+lnx`
...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=(sin^(-1)x)/(sqrt(1-x^2)) , show that (1-x^2)(d^2y)/(dx^2)-3x(dy)...

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

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  3. If y=x^x , prove that (d^2y)/(dx^2)-1/y((dy)/(dx))^2-y/x=0

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  4. Find (d^2y)/(dx^2) , if x=a t^2,\ \ y=2a t .

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  5. If x=acos^3theta and y=asin^3theta, then find the value of (d^2y)/(...

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  6. If y=asint and x=a(cost+logtant/2) , find (d^2y)/(dx^2) .

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  7. lf x=acostheta+bsintheta, y=asintheta-bcostheta then show that y^2(d^...

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  8. If x=sint,y=sinpt, prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2 y=...

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  9. If x=sintheta , y=cosptheta , prove that (1-x^2)y2-x y1+p^2y=0 , where...

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  10. If (x-a)^2+(y-b)^2=c^2, for some c >0, p rov et h a t([1+((dy)/(dx))^...

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  11. If y^2=a^2cos^2x+b^2sin^2x then prove that (d^2y)/(dx^2)+y=(a^2b^2)/(y...

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  12. If f(x)=|x|^3 , show that f"(x) exists for all real x and find it.

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  13. In (dy)/(dx),\ x is independent variable and y is the dependent va...

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  14. In (dy)/(dx),x is a independent variable and y is the dependent variab...

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  15. Find second order derivative of x^3+tanx

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  16. Find second order derivative of sin(logx)

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  17. Find second order derivative of log(sinx)

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  18. Find second order derivative of e^xsin5x

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  19. Find second order derivative of e^(6x)cos3x

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  20. Find second order derivative of x^3logx

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