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A function f is defined such that for al...

A function f is defined such that for all real `x, y` (a) `f(x+y)=f(x).f(y)` (b) `f(x)=1+xg(x)` where `lim_(x->0) g(x)=1.` prove that `f;(x)=f(x)` and `f(x)=e^x`

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A DAS GUPTA-Differentiation-Exercise
  1. Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)...

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

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

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

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

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

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

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  9. Prove that the derivative of a periodic function of period T is a peri...

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  10. Let f(x) be a function satisfying the condition f(-x) = f(x) for all r...

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  11. A function f:R->R satisfies the relation f((x+y)/3)=1/3|f(x)+f(y)+f(0)...

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  12. Find the sum of series sum(r=1)^nr.x^(r-1), using calculus.

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  13. Differential coefficient of log2(log2x)w.r.t x is

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  14. If f(x)=logx(logex) then f'(e)=

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  15. If x=a (cos t +log (tan ((t)/(2)) )) ,y =a sin t ,then (dy)/(dx) =

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  16. Let f'(x) = sin(x^2) and y = f(x^2 +1) then dy/dx at x=1 is

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  17. If x=cost , y=loget then at t=pi/2, (d^2y)/(dx^2)+((dy)/(dx))^2=

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  18. If f'(x)=sin x^2 and y=f(x^2+1) then dy/dx=

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  19. If x=cost , y=loget then at t=pi/2, (d^2y)/(dx^2)+((dy)/(dx))^2=

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  20. If tany = (2t)/(1-t^2) and sin x = (2t)/(1+t^2) then (dy)/(dx)=

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