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Differentiate x^(sin x), x gt 0 w.r.t x...

Differentiate `x^(sin x), x gt 0` w.r.t x

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The correct Answer is:
`x^(sin x-1).sin x + x^(sin x). cos x.log x`
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KUMAR PRAKASHAN-CONTINUITY AND DIFFERENTIABILITY-Textbook Illustrations for Practice Work
  1. Differentiate sqrt((x-3) (x^(2) + 4))/sqrt((3x^(2) + 4x + 5)) w.r.t x.

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  2. Differentiate a^(x) w.r.t x, where a is a positive constant

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  3. Differentiate x^(sin x), x gt 0 w.r.t x

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  4. Find (dy)/(dx), If y^(x) + x^(y) + x^(x)= a^(b)

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  5. Find (dy)/(dx), if x = a cos theta, y= a sin theta

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  6. Find (dy)/(dx), if x= a t^(2), y= 2 a t

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  7. Find (dy)/(dx), if x= a (theta + sin theta), y= a (1- cos theta)

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  8. Find (dy)/(dx), if x^((2)/(3)) + y^((2)/(3))= a^((2)/(3))

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  9. Find (d^(2)y)/(dx^(2)), if y= x^(3) + tan x

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  10. If y= A sin x + B cos x, then prove that (d^(2)y)/(dx^(2)) +y= 0

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  11. If y= 3e^(2x) + 2e^(3x), prove that (d^(2)y)/(dx^(2))-5 (dy)/(dx) + 6y...

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  12. If y= sin^(-1)x, show that (1 -x^(2)) (d^(2)y)/(dx^(2))-x (dy)/(dx)=0

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  13. Verify Rolle's theorem for the function y= x^(2) + 2, a = - 2 and b= 2

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  14. Verify Mean value Theorem for the function f(x)= x^(2) in the interval...

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  15. Differentiate w.r.t x, the following function: (i) sqrt(3x+2) + (1)...

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  16. Differentiate the following w.r.t x. (i) cos^(-1) (sin x) (ii) tan^...

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  17. Find f'(x) if f(x)= (sin x)^(sin x) for all 0 lt x lt pi.

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  18. For a positive constant a find (dy)/(dx), where y= a^(t + (1)/(t)), an...

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  19. Differentiate sin^(2)x w.r.t e^(cos x)

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  20. Examine the continuity of the function f(x)= 2x + 3 " at " x=1

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