Find the derivative using the chain rule and the quotient rule 
$$f(x) = \left(\frac{x}{x+1}\right)^4$$ Find $f'(x)$.

Here is my work:
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4\left(x+1\right)^3x^4}{\left(x+1\right)^8}$$
$$f'(x) = \frac{4x^3\left(x+1\right)^4-4x^4\left(x+1\right)^3}{\left(x+1\right)^8}$$
I know the final simplified answer to be:
$${4x^3\over (x+1)^5}$$
How do I get to the final answer from my last step? Or have I done something wrong in my own work?

First use the chain rule
$$(4*x^3)/(x+1)^3$$........()
Now derivate the inner function finding using quotient rule
$$((X+1)-x)/(x+1)^{2}=> 1/(x+1)^{2}$$.........(${*}{*}$)
Now multiply them both () &. (**) to get
&$${4x^{3}}/{(x+1)^{5}}$$
