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If y=x sin y then prove that xdy/dx=(y)/...

If y=x sin y then prove that `xdy/dx=(y)/(1-x cos y)`

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To prove that \( x \frac{dy}{dx} = \frac{y}{1 - x \cos y} \) given that \( y = x \sin y \), we will differentiate the equation and manipulate it step by step. ### Step-by-Step Solution: 1. **Start with the given equation:** \[ y = x \sin y \] ...
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NAGEEN PRAKASHAN ENGLISH-Continuity and Differentiability-Miscellaneous Exercise
  1. If y=x sin y then prove that xdy/dx=(y)/(1-x cos y)

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  2. Differentiate w.r.t. x the function. (3x^2-9x+5)^9

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  3. Differentiate w.r.t. x the functionsin^3x+cos^6x

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  4. y=(5x)^(3cos 2x) then find (dy)/(dx).

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  5. sin^(-1)(xsqrt(x)),0 le x le 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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