$$C=\frac{K\left(\sqrt{1-\frac{a^2}{b^2}}\right)}{K\left( \frac{a}{b} \right)}$$
where $K$ is the complete elliptic integral of the first kind.
I need to differentiate $C$ w.r.t. to $a$.
This occurs in a formula to calculate the capacitance of an interdigitated sensor. $a$ is the gap between the electrodes and I want to optimise the capacitance based on the size of this gap.
I'm guessing its too simple to let the derivative cancel the integral and I am left with the quotient of moduli above?