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A function f: R rarr R satisfies the equ...

A function `f: R rarr R` satisfies the equation `f(x+ y)= f(x).f(y) "for all" x, y in R, f(x) ne 0`. Suppose that the function is differentiable at x=0 and f'(0)=2, then prove that f'(x)= 2f(x)

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The correct Answer is:
`f'(x)= 2 f(x)`
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KUMAR PRAKASHAN-CONTINUITY AND DIFFERENTIABILITY-NCERT Exemplar Problems and Solution (Short Answer Type Questions)
  1. Examine the continuity of the function f(x)= {(1+x",","if " x le 2),(5...

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  2. Show that f(x) = |x-5| is continuous but not differentiable at x= 5

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  3. A function f: R rarr R satisfies the equation f(x+ y)= f(x).f(y) "for ...

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  4. Differentiate 2^(cos^(2)x)

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  5. Differentiate (8^(x))/(x^(8))

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  6. Differentiate log (x + sqrt(x^(2) + a))

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  7. Differentiate the following with respect to x: log [log (log x^(5))]

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  8. Differentiate sin sqrtx + cos^(2) sqrtx

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  9. Differentiate sin^(n) (a x^(2) + bx + c)

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  10. Differentiate cos (tan sqrt(x+ 1))

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  11. Differentiate sin x^(2) + sin^(2)x + sin^(2) (x^(2))

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  12. Differentiate sin^(-1) ((1)/(sqrt(x + 1)))

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  13. Differentiate (sin x)^(cos x)

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  14. Differentiate sin^(m) x.cos^(n) x

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  15. Differentiate (x+1)^(2) (x + 2)^(3) (x + 3)^(4)

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  16. Differentiate cos^(-1) ((sin x + cos x)/(sqrt2)), - (pi)/(4) lt x lt (...

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  17. Differentiate tan^(-1) sqrt((1-cos x)/(1 + cos x)), - (pi)/(4) lt x lt...

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  18. Differentiate tan^(-1) (sec x + tan x), -(pi)/(2) lt x lt (pi)/(2)

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  19. Differentiate, tan^(-1) ((a cos x- b sin x)/(b cos x + a sin x)), (-pi...

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  20. Differentiate sec^(-1) ((1)/(4x^(3)-3x)), 0 lt x lt (1)/(sqrt2)

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