Bound on derivative of $e^{\frac{-\delta^2}{\delta^2-\left( \delta-x \right)^2}+1}$

Question

Define $$f(x):=e^{\frac{-\delta^2}{\delta^2-\left( \delta-x \right)^2}+1}$$ for $x\in (0,\delta) $ where $0.5 > \delta>0$.

is it possible to get a bound on the $L^\infty$ norm of $f'$ on the domain $(0,\delta)$.

I tried writing $f(x) = e^{g(x)}$ and $f'(x) = g'(x)e^{g(x)}$, but it seems hard this way.

Answer 1

$$\lim_{x\to 0} f'(x)=0=\lim_{x\to\delta} f'(x)$$ Therefore, $$ \tilde{f}(x)=\begin{cases} f'(x) & x\in(0,\delta)\\0 & x=0~or~x=\delta\end{cases} $$ is continuous on $[0,\delta]$. Further $[0,\delta]$ is compact which implies that $\tilde f$ is bounded on $[0,\delta]$. Therefore $f'=\tilde f\mid_{(0,\delta)}$ is bounded on $(0,\delta)$.