Question

how to find 50th derivative of a function ?

Answer 1

 Take the derivative a few times and look for a pattern. 

For example, we can find the derivative of f(x) = e^(3x) this way: 

f(x) = e^(3x) 
f ' (x) = 3e^(3x) 
f '' (x) = 9e^(3x) 
f ''' (x) = 27e^(3x) 
f^(4) (x) = 81e^(3x) 

It looks like the pattern is that, each time we take the derivative, we multiply the function we had before by 3. 

Multiplying f(x) by 3 fifty times gives (3^50)e^(3x). 

So the 50th derivative of e^(3x) is (3^50)e^(3x). 

Answer 2

we find the 1st derivative , then the second derivative, and so on...
We could find a pattern of the n th derivative.

f(x) = a x^n            a and n are constants.

if n < 50,    f ⁵°  = 0
if n = 50  ,  f ⁵°  = 50! * a
if n > 50,    f ⁵°  = n (n-1)..(n-49) a x^(n-50)
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f(x) = a Sin bx

f ' = a b cos bx
f '' = - a b² Sin bx = - b² f
f ³ = - a b³ Cos bx  = - b² f '
f ⁴ = a b⁴ Sin bx  = - b² f '' = b⁴ f
f ⁵ =  a b⁵ Cos bx = - b² f ³  =
f ⁶ = - a b⁶ Sin bx  = - b² f ⁴ = - b⁶ f
f ⁷ = - a b⁷ Cos bx = - b² f ⁵
f ⁸ = a b⁸ Sin bx  = b⁸ f

so  f ⁴⁸ =  b⁴⁸ f
  f ⁴⁹ = b⁴⁹ a Cos bx
  f ⁵° = - b⁵° a Sin bx = - b⁵° f
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f(x) = Log ax

 f ' = a x⁻¹          f '' = - a x⁻²            f ³ = 2 a x⁻³        f ⁴ = - 3! x⁻⁴
so  f ⁵° = - 49! x⁻⁵°