The question is: Differentiate the following function, identifying any general rules of calculus that you use and simplify your answer as far as possible.

\(\displaystyle f(x)=\frac{e^{tanx}}{(x^3+1)}\)

Rules: quotient rule, product rule, sum/multiple rules

forulma: \(\displaystyle f'(x)=\frac{g(x)e'(x)-e(x)g'(x)}{(g(x))^2}\)

\(\displaystyle e(x)=e^{tanx}\), \(\displaystyle u=tanx\)

\(\displaystyle e'(u)=e^u\), \(\displaystyle \frac{du}{dx}=sec^2x\)

\(\displaystyle \frac{de}{du}=e^u\)

\(\displaystyle \frac{de}{dx}=e^u x sec^2x=sec^2 x e^{tanx}\)

\(\displaystyle g(x)=(x^3+1)^2\), \(\displaystyle u=x^3+1\)

\(\displaystyle g'(u)=u^2\), \(\displaystyle \frac{du}{dx}=3x^2\)

\(\displaystyle \frac{dg}{du}=2u\)

\(\displaystyle \frac{dg}{dx}=2(x^3+1)3x^2\)

\(\displaystyle f'(x)=\frac{(x^3+1)^2 x (sec^2x e^{tanx})-(e^{tanx})x(2(x^3+1)3x^2)}{((x^3+1)^2)^2}\)

\(\displaystyle f'(x)=\frac{(x^3+1)^2 x sec^2x e^{tanx}-e^{tanx}x2(x^3+1)3x^2}{(x^3+1)^4}\)

I hope this is right so far, but am not sure how to simplify any further (unless I've done something wrong?) (Worried)