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If x=a(t+1/t) and y=a(t-1/t) , prove tha...

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

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To prove that \(\frac{dy}{dx} = \frac{x}{y}\) given \(x = a\left(t + \frac{1}{t}\right)\) and \(y = a\left(t - \frac{1}{t}\right)\), we will follow these steps: ### Step 1: Differentiate \(x\) and \(y\) with respect to \(t\) 1. **Differentiate \(x\)**: \[ x = a\left(t + \frac{1}{t}\right) \] ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If x=e^(cos2t) and y=e^(sin2t) , prove that (dy)/(dx)=-(ylogx)/(xlogy)

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  2. If x=cost and y=sint , prove that (dy)/(dx)=1/(sqrt(3)) at t=(2pi)/3

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

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  4. If x=sin^(-1)((2t)/(1+t^2)) and y=tan^(-1)((2t)/(1-t^2)) , -1<t<1 , pr...

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  5. If x and y are connected parametrically by the equations given, wit...

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  6. If a gt0, x=(t+(1)/(t))^(a) and y=a^((t+(1)/(t))), find (dy)/(dx).

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  7. If x=a((1+t^2)/(1-t^2)) and y=(2t)/(1-t^2) , find (dy)/(dx)

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  8. If x=10(t-sint) , y=12(1-cost) , find (dy)/(dx) .

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  9. If x=a(theta-sintheta) and, y=a(1+costheta) , find (dy)/(dx) at theta=...

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

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  11. If x=cost(3-2\ cos^2t) and y=sint(3-2s in^2t) find the value of (dy)/(...

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

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  13. If x=3sint-sin3t ,y=3cos t-cos3t ,"f i n d"(dy)/(dx)"a t"t=pi/3dot

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

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  15. Differentiate logsinx with respect to sqrt(cosx) .

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  16. Differentiate tan^(-1)((1+2x)/(1-2x)) with respect to sqrt(1+4x^2) .

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  17. Differentiate tan^(-1)((sqrt(1+x^2)-1)/x) with respect to tan^(-1)x ,x...

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  18. Differentiate sin^(-1)((2x)/(1+x^2)) with respect to tan^(-1)x ,-1ltxl...

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  19. Differentiate x^x with respect to xlogx .

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  20. Differentiate tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x...

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