first of all, excuse me for my English
I must calculate the inverse function of $$f(x)=-x^3-x+6$$ and then the derivative. my problem isn't the derivative, my problem is that I just can't calculate the inverse for third order polynomials.
I begin by doing the following:
$y=f(x)=-x^3-x+6 \rightarrow y-6=-x^3-x \rightarrow -y+6=x^3+x$
and then I just don't know what to do
thank you for the help
Consider the cubic equation $$-x^3-x+(6-y)=0$$ Its discriminant $$\Delta=-27 (y-6)^2-4$$ is negative for all $y$; then, only one real root and you can compute its inverse.
Using the hyperbolic method $$x=-\frac{2}{\sqrt{3}}\,\sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{3\sqrt{3}}{2} (y-6)\right)\right)$$ looks pretty nice