Calculating differential of inverse function.

Question

trying to find $(f'^{-1})(a)$ and am getting the wrong answer.

Answer 1

You only need to find the value of the inverse of the derivative at a single point, so that makes things easier. You are right to start with the canonical formula $$(f')^{-1}(a)=\frac{1}{f'(f^{-1}(a))}$$ So let's compute the ingredients. Now, $f(x)=\frac{3}{x+2}$, correct? So, its derivative is $f'(x)=-\frac{3}{(x+2)^{2}}$. Next, set up the equation $$a=\frac{3}{x+2}$$ Solve for $x$ in terms of $a$, and this will give us $f^{-1}(a)$: $$a(x+2)=3$$ $$ax=3-2a$$ $$x=\frac{3-2a}{a}=\frac{3}{a}-2$$ so $$f^{-1}(a)=\frac{3}{a}-2$$ Thus, $$f'(f^{-1}(a))=-\frac{3}{(\frac{3}{a}-2+2)^{2}}=-\frac{3}{(\frac{3}{a})^{2}}=-\frac{3}{\frac{9}{a^{2}}}=-\frac{a^{2}}{3}$$ Thus, $$(f')^{-1}(a)=-\frac{3}{a^{2}}$$