Question

So, I am having trouble with this homework problem on higher order derivatives... can anyone point me in the right direction? I can get through the first and maybe second derivation, but not all the way to the fourth.
Find f''''(x) for f(x) = -5x^4/(1-x)
Note: There is a way of doing this problem without using the quotient rule 4 times.

Answer 1

The only way I personally know of is to find the derivative 4 times.
Find the derivative of the first, then take that answer and find the derivative of that to have the second derivative, then take the answer of f''(x), and take the derivative, which would be F'''(x), then take the derivative of that and you would have your final answer f''''(x). Would you like me to draw out the whole process?

Answer 2

the first being f(x)= -5x^4/(1-x)

Answer 3

\[-\frac{5 x^4}{1-x}=5 x^3+5 x^2+5 x+\frac{5}{x-1}+5 \]The RHS of the above came from applying the Mathematica function, Apart, to the LHS.

I presume that the application of partial fraction decomposition by hand would arrive at the same result.

The fourth derivative of\[5 x^3+5 x^2+5 x+\frac{5}{x-1}+5 \] is\[\frac{120}{(x-1)^5} \]