How to find the second derivative of the inverse function of $f(x)$ at $x=0$?

Question

If $f\displaystyle(x)=\int _{ \sin x }^{ \cos x }{ \frac { dt }{ e^{ t }\sqrt { 1-t^2 } } } $ where $x\in[0,\pi/2]$ then how to find the second derivative of the inverse function of $f(x)$ at $x=0$ ?

Answer 1

By substituting $t = \sin u$ or $u = \arcsin t$, one sees that $$f(x) = \int_x^{{\pi \over 2} - x} e^{-\sin u}\,du$$ Taking derivatives using the Fundamental Theorem of Calculus, one obtains $$f'(x) = -e^{-\sin({\pi \over 2} - x)} - e^{-\sin x}$$ $$= -e^{-\cos x} - e^{-\sin x}$$ Taking a second derivative gives $$f''(x) = -e^{-\cos x}\sin x + e^{-\sin x} \cos x$$

The formula for the second derivative of an inverse function was answered here second derivative of the inverse function

So if $g(x)$ is the inverse of $f$ we have $$g''(f(x)) = - {f''(x) \over (f'(x))^3}$$ Since $f({\pi \over 4}) = 0$ here, we have $$g''(0) = -{f''({\pi \over 4}) \over (f'({\pi \over 4}))^3}$$ Plugging back in we get $${{\sqrt 2 \over 2}(e^{-{\sqrt 2 \over 2}} - e^{-{\sqrt 2 \over 2}}) \over (-e^{-{\sqrt 2 \over 2}} - e^{-{\sqrt 2 \over 2}})^{3}}$$ $$= 0$$

Answer 2

Denote the inverse function by $h(x)$ so that $h(f(x)) = x$. Using implicit differentiation, we have

$$ h'(f(x)) \cdot f'(x) = 1, \\ h''(f(x)) \cdot (f'(x))^2 + h'(f(x)) \cdot f''(x) = h''(f(x)) \cdot (f'(x))^2 + \frac{f''(x)}{f'(x)} = 0.$$

Note that $f(\frac{\pi}{4}) = 0$ so

$$ h''(0) = -\frac{f''(\frac{\pi}{4})}{(f'(\frac{\pi}{4}))^3}. $$

Calculate $f'(\frac{\pi}{4})$ and $f''(\frac{\pi}{4})$ using the fundamental theorem of calculus and you're done.

---

The function $f$ is of the form $$ f(x) = \int_{g(x)}^{r(x)} s(t) \, dt = \int_q^{r(x)} s(t) \, dt - \int_q^{g(x)} s(t) \, dt$$ where $q \in \mathbb{R}$ is some arbitrary point. Using the chain rule and the fundamental theorem of calculus, we see that $$ f'(x) = s(r(x)) \cdot r'(x) - s(g(x)) \cdot g'(x). $$

Applying this to your specific $f$, we have

$$ f'(x) = \frac{-\sin(x)}{e^{\cos(x)}\sqrt{1 - \cos^2(x)}} - \frac{\cos(x)}{e^{\sin(x)} \sqrt{1 - \sin^2(x)}} = - \left( \frac{1}{e^{\cos(x)}} + \frac{1}{e^{\sin(x)}} \right) = -e^{-\cos(x)} - e^{-\sin(x)}. $$