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If y=(sinx-cosx)^(sinx-cosx),pi/4<x<(3pi...

If `y=(sinx-cosx)^(sinx-cosx),pi/4

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Let `y=(sinx-cosx)(sinx-cosx)`
`implieslogy=log(sinx-cosx)^((sinx-cosx))`
`=(sinx-cosx)log(sinx-cosx)`
Differentiate both sides w.r.t. x
`(1)/(y)(dy)/(dx)=(sin x-cos x)(d)/(dx)log(sinx-cosx)+log(sinx-cosx)(d)/(dx)(sinx-cosx)`
`implies(dy)/(dx)=[((sin x-cos x))/(sinx-cosx)*(d)/(dx)(sinx-cosx)+log(sinx-cosx)*(cosx+sinx)]`
`=y[(cosx+sinx)+(cosx+sinx)log(sinx-cosx)]`
`=(sinx-cosx)^((sinx-cosx))(cosx+sinx)[1+log(sinx-cosx)]`
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NAGEEN PRAKASHAN ENGLISH-Continuity and Differentiability-Miscellaneous Exercise
  1. sin^(-1)(xsqrt(x)),0 le x le 1

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  2. Differentiate w.r.t. x the function (cos^(-1)x/2)/(sqrt(2x+7)),-2<x<2

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  3. Prove that:cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-s...

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  4. (log x)^(log x),x gt1

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  5. Differentiate w.r.t. x the function cos" "(a" "cos" "x" "+" "b" "s in"...

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  6. If y=(sinx-cosx)^(sinx-cosx),pi/4<x<(3pi)/4,"f i n d"(dy)/(dx)dot

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  7. Differentiate w.r.t. x the function x^x+x^a+a^x+a^a ,for some fixed a...

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  8. Differentiate x^x^(2-3)+(x-3)^x^2 with respect to x :

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  9. Find (dy)/(dx),if y=12(1-cost), x=10(t-sint)

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  10. If y=sin^(-1)x+sin^(-1)sqrt(1-x^2) , find (dy)/(dx) in each of the ...

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  11. "If "xsqrt(1+y)+ysqrt(1+x)=0," prove that "(dy)/(dx)=-(1)/((x+1)^(2)).

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  12. "If "(x-a)^(2)+(y-b)^(2)=c^(2), for some cgt0, prove that ([1+((dy)/(d...

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  13. "If "cos y = x cos(a+y)," with "cos a ne pm1," prove that "(dy)/(dx)=(...

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  14. If x=a(cost+tsint) and y=a(sint-tcost), find (d^2y)/(dx^2)

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

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  16. Using mathematical induction prove that d/(dx)(x^n)=n x^(n-1) for all...

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  17. Using the fact that s in (A + B) = s in A cos B + cos A s in Band the ...

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  18. Does there exist a function which is continuous everywhere but not ...

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  19. If y=|f(x)g(x)h(x)l m n a b c| , prove that (dy)/(dx)=|f^(prime)(x)g^(...

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  20. "If "y=e^(a cos^(-1)x),-1lexle1," show that "(1-x^(2))(d^(2)y)/(dx^(2)...

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