Answer
$\frac{2}{\pi}$
Work Step by Step
We have a formula $f(x)$, and are looking for the derivative of its inverse formula, $h(x)$.
The formula for the derivative of $h(x)$ evaluated at a specific point, (i.e. where a=3) is as follows
$h'(a)=\frac{1}{f'(h(a))}$
The first step is to find $f'(x)$ respectively. By simple differentiation and the chain rule we get that
$f'(x)=2x+\frac{\pi sec(\frac{\pi x}{2})tan(\frac{\pi x}{2})}{2}$
Secondly, we will use the a-value to find the x-value in $f(x)$ that corresponds to it.
$3=3+x^{2}+tan(\frac{\pi x}{2})$
By observing, we notice that the only way that 3 could equal the other side is if the x-terms equal 0, and since x is multiplying in both of them, the x-value that satisfies the equation is 0.
$3=3+0^{2}+tan(\frac{\ 0\times \pi}{2})$
$3=3+0+tan(0)$
$3=3+0+0$
$3=3$
Thus, we can now state that $h(3)=0$, since $f(0)=3$
$h'(3)=\frac{1}{f'(h(3))}$
$h'(3)=\frac{1}{f'(0)}$
To continue working with the formula, we must solve for $f'(0)$
$f'(0)=2(0)+\frac{\pi sec(\frac{\pi (0)}{2})tan(\frac{\pi (0)}{2})}{2}$
$f'(0)=0+\frac{\pi sec(0)tan(0)}{2}$
$f'(0)=0+\frac{\pi (1)}{2}$
$f'(0)=\frac{\pi}{2}$
If plugged into the equation,
$h'(3)=\frac{1}{f'(0)}$
$h'(3)=\frac{1}{\frac{\pi}{2}}$
Thus resulting as
$h'(3)=\frac{2}{\pi}$
