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If e^x+e^y=e^(x+y),prove that (dy)/(dx)+...

If `e^x+e^y=e^(x+y)`,prove that `(dy)/(dx)+e^(y-x)=0`

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To prove that \(\frac{dy}{dx} + e^{y-x} = 0\) given that \(e^x + e^y = e^{x+y}\), we can follow these steps: ### Step 1: Start with the given equation We have: \[ e^x + e^y = e^{x+y} \] ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If (sinx)^y=(cosy)^x ,prove that (dy)/(dx)=(logcosy-ycotx)/(logsinx+xt...

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  2. If y=cos^(-1){(2x-3sqrt(1-x^2))/(sqrt(13))},"f i n d"(dy)/(dx)

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  3. If e^x+e^y=e^(x+y),prove that (dy)/(dx)+e^(y-x)=0

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  4. If xsin(a+y)+sina cos(a+y)=0, prove that (dy)/(dx)=sin^2(a+y)/sina

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  5. Find (dy)/(dx) in each of the following: (x^2+y^2)^2=x y

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  7. Differentiate the following functions with respect to e^(sin^{-1}{...

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  8. Differentiate the following functions with respect to "sin"(2sin^(-...

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  9. Differentiate the following function sqrt(tan^(-1)(sqrtx))

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

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  11. Differentiate tan^(-1)((1+a x)/(1-a x)) with respect to sqrt(1+a...

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  12. Differentiate tan^(-1)((x-1)/(x+1)) with respect to sin^(-1)(3x-...

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  13. If y = (logcos x sinx)(logsin x cos x)–1 + sin–1(2x/(1 + x2)) then pro...

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  14. If y=sec^(-1)((x+1)/(x-1))+sin^(-1)((x-1)/(x+1)),x > 0. Find (dy)/(dx)...

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  15. Differentiate tan^(-1)(x/(sqrt(1-x^2))) with respect to sin^(...

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  16. Differentiate sin^(-1)sqrt(1-x^2) with respect to . if 0<x<1.

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  17. If ysqrt(x^2+1)=log(sqrt(x^2+)1-x),show that" " (x^2+1)(dy)/(dx)+x y+1...

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  18. If e^x+e^y=e^(x+y) , prove that (dy)/(dx)=-(e^x(e^y-1))/(e^y(e^x-1)) o...

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  19. If y=xsin(a+y),prove that (dy)/(dx)=(sin^2(a+y))/(sin(a+y)-ycos(a+y))d...

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  20. If x ylog(x+y)=1,prove that(dy)/(dx)=-(y(x^2y+x+y))/(x(x y^2+x+y))dot

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