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If x^(y) = e^(2(x-y)), " then" f'(x) i...

If `x^(y) = e^(2(x-y)), " then" f'(x) ` is equal to

A

`(2(log x + 1))/((log x + 2)^(2))`

B

`((log x + 1)^(2))/((log x + 2))`

C

`(2log x )/(log x + 2)`

D

`(2(log x+1))/((log x + 2))`

Text Solution

Verified by Experts

The correct Answer is:
A
`x^(y) = e^(2 (x -y)) `
` therefore y log x = 2 (x -y)`
` rArr y (log x + 2 ) = 2x `
`rArr y = (2x)/(log x + 2)`
` rArr (dy)/(dx) = ((log x + 2 ) (2) - 2x.(1)/(x))/((log x + 2 )^(2))`
`= (2 log x + 4 - 2)/(( log x + 2)^(2)) = (2(log x + 1))/((log x + 2)^(2))` .
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DIFFERENTIATION -EXERCISE 2 (MISCELLANEOUS PROBLEMS)
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  2. If f(x)=cot^(-1)((x^(x)-x^(-x))/(2)) then f'(1) equals

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  3. If x^(y) = e^(2(x-y)), " then" f'(x) is equal to

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  4. If sqrt(1-x^(2n))+sqrt(1-y^(2n))=a^n(x^n-y^n), prove that y^(n-1)*sqr...

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  5. If f(x)=sin^(-1)(sinx)+cos^(-1)(sinx) and phi(x)=f(f(f(x))), then phi'...

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  6. Let x^cosy + y^cosx = 5, then

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  7. If y = sin x^(@) " and " u = cos x " then " (dy)/(dx) is equal to

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  8. If y=tan^(-1)1/(1+x+x^2)+tan^(-1)1/(x^2+3x+3)+tan^(-1)1/(x^2+5x+7)++ u...

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  9. If y=tan^(-1)((sinx+cosx)/(cosx-sinx)) , then (dy)/(dx) is equal to

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  10. If y = cos^(-1) ((2cos x - 3 sin x)/(sqrt(13))), " then" (dy)/(dx) i...

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  11. Let y=(x^(8)+x^(4)+1)/(x^(4)+x^(2)+1)."If"(dy)/(dx)=ax^(3)+bx. Then,

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  12. If y=sin^(-1){(5x+12 sqrt(1-x^(2)))/(13)}, find (dy)/(dx).

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  13. If y=sqrt(x+sqrt(y+sqrt(x+sqrt(y+...oo)))), then (dy)/(dx) is equal to

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  14. if y=sin^-1[sqrt(x-ax)-sqrt(a-ax)] then prove that 1/(2sqrtx sqrt(1-x)...

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  15. If y=log(x^(2)+4)(7x^(2)-5x+1), then (dy)/(dx) is equal to

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  16. If x=sectheta-costheta and y=sec^n theta- cos^n theta then show that (...

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  17. If x^(2) + y^(2) = a^(2) , then a is equal to

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  18. Let f(x) be a polynomial.Then, the second order derivative of f(e^x) i...

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  19. If y = sin^(n) x cos nx, " then" (dy)/(dx) is

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  20. If y = sec^(-1) [ "cosec x"] + "cosec"^(-1) [sec x] + sin^(-1) [cos x...

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