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Prove that d/(dx)|(u1,v1,w1),(u2,v2,w2),...

Prove that `d/(dx)|(u_1,v_1,w_1),(u_2,v_2,w_2),(u_3,v_3,w_3)|=|(u_1,v_1,w_1),(u_2,v_2,w_2),(u_4,v_4,w_4)|` where `u,v,w` are functions of `x and (du)/(dx)=u_1,(d^2u)/(dx^2)=u_2,` etc.

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A DAS GUPTA-Differentiation-Exercise
  1. Find (dx)/(dt) when x = sin^-1(t.sqrt(1-t) +sqrt(t) sqrt(1-t^2)).

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

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

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

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

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

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

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

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

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

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

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  15. A function f is defined such that for all real x, y (a) f(x+y)=f(x).f(...

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  16. A function f is defined such that for all real x, y (a) f(x+y)=f(x).f(...

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  17. Let f(x+y)=f(x)+f(y) for all real x,y and f'(0) exists. Prove that f'(...

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  18. Let f((x+y)/(2))=1/2 |f(x) +f(y)|for all real x and y, if f '(0) exist...

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  19. A function f(x) is so defined that for all real x,{f(x)}^n = f(nx). Pr...

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  20. Prove that the derivative of (a) an odd function is an even function;

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