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If y=sqrt(x(log)e x) , then find (dy)/(d...

If `y=sqrt(x(log)_e x)` , then find `(dy)/(dx)` at `x=e` .

A

`(1)/(e)`

B

`sqrt(e)`

C

`(1)/(sqrt(e))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
Given , ` y = sqrt(x log_(e) x)`
` therefore y^(2) = x* log x `
` 2y (dy)/(dx) = (x)/(x) + log x `
` therefore (dy)/(dx) = (1)/(2 sqrt(x log x))[ 1 + log x]`
` therefore ((dy)/(dx))_(x = e) = (1)/(2sqrt(r)) [ 1 + 1]= (1)/(sqrt(e))`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DIFFERENTIATION -EXERCISE 2 (MISCELLANEOUS PROBLEMS)
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  2. If xe^(xy)=y+sin^2x then at x=0 (dy)/dx=

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  3. If y=sqrt(x(log)e x) , then find (dy)/(dx) at x=e .

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  4. Let y=e^(2x)dotT h e n((d^2y)/(dx^2))((d^2x)/(dy^2))i s

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  5. If f(x)=|x-2| and g(x)=f(f(x)), then g'(x)"for "x gt 20,is

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  6. If y=(a+b x^(3/2))/(x^(5/4))a n dy^(prime)=0a tx=5, then the value o...

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  7. The value of (d)/(dx) (|x-1| + |x -5|) at x = 3 is

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  8. If y=f(x)" and "ycosx+cosy=pi, then the value of f''(0) is

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  9. Derivatives of y = cos^(-1) sqrt((cos 3x)/(cos ^(3) x)) with respect ...

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  10. if sqrt(x^2+y^2)=ae^(tan^-1 (y/x)) , a > 0, (y(0) > 0) then y"(0) equa...

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  11. If y = In ((x)/(a + bx))^(x), "then " x^(3)(d^(y))/(dx^(2)) is equa...

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  12. If (d^(2)x)/(dy^(2)) ((dy)/(dx))^(3) + (d^(2) y)/(dx^(2)) = k , then...

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  13. If f(x) = |cos x - sin x| , " then " '((pi)/(6)) is equal to

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  14. If f' (x) = sin (log x) and y = f((2x+3)/(3- 2x)), " then " (dy)/(...

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  15. If u = x^(2) + y^(2) " and " x = s + 3t, y = 2s - t , " then " (d^(2...

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  16. If the function f(x) is defined by f(x) = a + bx and f^(r) = fff … ...

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  17. If f(x) = x^(3) + x^(2) * f' (1)+ xf''(2) + f''(2) + f'''(3) , AA x i...

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  18. If x = tan""(y)/(2) - log[((1 + tan""(y)/(2))^(2))/(tan""(y)/(2))], "t...

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  19. If y=sec^-1([sqrtx+1]/[sqrtx-1])+sin^-1([sqrtx-1]/[sqrtx+1]), then dy/...

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  20. If y=(a^cos^((-1)x))/(1+a^cos^((-1)x)) and z=a^cos^((-1)x) , then (dy)...

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