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(sinx)^(cosx)...

`(sinx)^(cosx)`

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Let `y = (sinx)^(cosx)`
`rArr logy = log(sinx)^(cosx)= cosxlogsinx`
`:. (d)/(dy) logy.(dy)/(dx)=(d)/(dx)(cosx.logsinx)`
`rArr (1)/(y). (dy)/(dx) = cosx . (d)/(dx) log sinx + logsinx.(d)/(dx) cosx`
`= cosx. (1)/(sinx).(d)/(dx)sinx+logsinx.(-sinx)`
`= cotx. cosx - log(sinx).sinx, [:' cotx = (cosx)/(sinx)]`
`:. (dy)/(dx) = y[(cos^(2)x)/(sinx)-sinx.log(sinx)]`
`= sinx^(cosx)[(cos^(2)x)/(sinx)-sinx.log(sinx)]`
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NCERT EXEMPLAR-CONTINUITY AND DIFFERENTIABILITY-Continuity And Differentiability
  1. Differentiate sinx^(2)+sin^(2)x+sin^(2)(x^(2))

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  2. Differentiate sin^(-1)\ (1)/(sqrt(x+1))

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  3. (sinx)^(cosx)

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  4. Differentiate sin^(m)x*cos^(n)x

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  5. Differentiate (x+1)^(2)(x+2)^(3)(x+3)^(4)

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  6. Simplify: cos^(-1)((sinx+cosx)/(sqrt(2))),\ \ pi/4<x<(5pi)/4

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  7. Differentiate tan^(-1){sqrt((1-cosx)/(1+cosx))},\ -pi<x<pi with respec...

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  8. Differentiate tan^(-1)(secx+tanx) , -pi/2<x<pi/2 with respect to x :

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  9. Differentiate the following functions with respect to x : tan^(-1)(...

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  10. Differentiate cos^(-1)(4x^3-3x) with respect to x , if x in (1/2,\ 1)

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  11. Differentiate tan^(-1)((3a^2x-x^3)/(a^3-3a x^2)),\ -1/(sqrt(3))<x/a<1/...

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  12. y=tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2))),w h e ...

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  13. If x=a(t+1/t) and y=a(t-1/t) , prove that (dy)/(dx)=x/y

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  14. Find (dy)/(dx) , when x=e^(theta)(theta+1/theta) and y=e^(-theta)(thet...

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  15. If x=3costheta-cos^3theta y=3sintheta-sin^3theta find dy/dx

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  16. If sinx=(2t)/(1+t^2) , tany=(2t)/(1-t^2) , find (dy)/(dx) .

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  17. If x=(1+logt)/(t^2),\ \ y=(3+2logt)/t ,\ \ find (dy)/(dx) .

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  18. If x=e^(cos2t) and y=e^(sin2t) , prove that (dy)/(dx)=-(ylogx)/(xlogy)

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  19. If x=asin2t(1+cos2t) and y=bcos2t(1-cos2t) , show that at t=pi/4 , (dy...

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  20. If x=3sint-sin3t , y=3cost-cos3t , find (dy)/(dx) at t=pi/3 .

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