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If x=f(t) and y=phi(t), prove that (d^2y...

If `x=f(t) and y=phi(t),` prove that `(d^2y)/(dx^2)=(f_1phi_2-f_2phi_1)/(f_1^3)` where suffixes denote differentiation `w.r.t.t.`

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A DAS GUPTA-Differentiation-Exercise
  1. Find the differentiation of log((1-sqrt(x)))(sin^-1(1-sqrt(x))) with r...

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  2. If y=sqrt((t-alpha)/(beta-t)) and x=sqrt((t-alpha)(beta-t)) then find ...

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  3. If x=f(t) and y=phi(t), prove that (d^2y)/(dx^2)=(f1phi2-f2phi1)/(f1^3...

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  4. If x=sectheta-costheta and y=sec^n theta- cos^n theta then show that (...

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  5. If x=costheta,ysin^3theta,"p r o v et h a t" y(d^2y)/(dx^2)+((dy)/(...

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  6. If x = 2 cos t - cos 2t , y = 2 sin t - sin 2t, then the value of ...

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  7. Find (dx)/(dt) when x = sin^-1(t.sqrt(1-t) +sqrt(t) sqrt(1-t^2)).

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  8. If x^(2)+y^(2)=t andx^(4)+y^(4)=t^(2)+(1)/(t^(2)), then prove that (dy...

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  9. Prove that d/(dx)|(u1,v1,w1),(u2,v2,w2),(u3,v3,w3)|=|(u1,v1,w1),(u2,v2...

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  10. If fr(x),gr(x),hr(x),r=1,2,3 are polynomials such that fr(a)=gr(a)=h ,...

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  11. If y = cos ax, prove that |(y,y1,y2),(y3,y4,y5),(y6,y7,y8)|=0 where yr...

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  12. Let Delta(x)=|(x^2-1,x+1,x-2),(2x^2-1,3x,3x-3),(x^2+4,2x-1,2x-1)|. Pro...

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  13. If y=sin(2sin^-1x)show that (1-x^2)(d^2y)/(dx^2)= xdy/dx-4y

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  14. If y=a cos (logx)+b sin (logx), prove that x^(2)y(2)+xy(1)+y=0.

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  15. If y=x^(n-1) log x, prove that (x^2y2)+(3-2n)xy1+(n-1)^2.y=0 where y1...

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  16. Let y=tan^-1sqrt(x^2-1). Prove that (2x^2-1)((dy)/(dx))+x(x^2-1)((d^2...

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  17. If y=(a x+b)/(x^2+c), prove that (2x y1+y)y3=3(x y2+y1)y2.

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  18. Let y=f(x).phi(x) and z=f'(x).phi'(x). prove that 1/y*(d^2y)/(dx^2)=1/...

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  19. Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)...

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  20. Let f'(x) exists for all x != 0 and f(xy) = f(x) + f(y) for all real x...

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