Create formula for n-th derivative of a certain function

Question

                  1

    If f(x) =   ------------     find f ⁽⁵⁰⁾ (x), that is the 50^th derivative of f(x)

                        2x – 3

Answer 1

f(x)=(2x-3)^-1

f'(x)=-1(2)(2x-3)^-2

f''(x)=-1(-2)(2)(2x-3)^-3

f'''(x)=-1(-2)(-3)(2)(2)(2)(2x-3)^-4

f^iv(x)=-1(-2)(-3)(-4)(2)(2)(2)(2)(2x-3)^-5

From the pattern

fⁿ(x)=(-1)ⁿ(n!)(2)ⁿ(2x-3)^-(n+1)

f⁵⁰(x) = 50!2⁵⁰(2x-3)^-51

Answer 2

No one expects to do 50 derivatives, so there must be a formula we can develop to represent any derivative of f(x).

We will write the first few derivatives, but will leave most of the numerator’s computations deliberately  undone.

                            -1 ^. 2

          f ‘(x) =    ---------      

                        (2x – 3)²

                      ^-2  ^.    -1 ^.  2²

          f ‘’(x) =  - ---------      

                        (2x – 3)³

                        -3  ^.   -2  ^.  -1 ^.  2³

          f ‘’’(x) =  ----- ---------      

                        (2x – 3)⁴

              

                           -4 ^. -3 ^.  -2 ^. -1 ^.  2⁴

          f ⁽⁴⁾ (x) =    --------------------  

                                  (2x – 3)⁵

                            -5 ^.  -4 ^. -3 ^.  -2 ^. -1 ^.  2⁵

          f ⁽⁵⁾ (x) =    --------------------  

                                  (2x – 3)⁶

We now have sufficient information to draw come conclusions

1)     The power of the denominator decreases by one with each higher order derivative

2)     Because of the chain rule, another factor of 2 will be generated as part of a higher order derivative

3)     Ignoring the “minus signs” for a moment, the numerator contains a factorial that starts with the same number as the power of 2

4)     The number of “minus signs” in each numerator matches the power of 2 in the numerator. This of course produces alternating signs in the calculated derivatives. The same effect can be accounted for by using a factor of (-1)ⁿ  in the numerator. Combine it with the power of 2 to make a simpler formula.

All of this is summarized by the formula:

  

                                 50! ^.  (-2)⁵⁰

          f ⁽⁵⁰⁾ (x) =    --------------------  

                                  (2x – 3)⁵¹