Derivative of Modified Bessel function of second kind

Question

I have to differentiate the following function with respect to $x \dfrac{dF}{dx}$, where $\alpha$ is constant: $$\\F(x)=\sqrt{4\alpha x}K_v(\sqrt{4\alpha x})$$

I am aware that chain rule of differentiation will apply but what about the Bessel function?

Any clue?

Answer 1

HINTS

Given a function $f$ where $f = f(ax)$ i.e. is a function of $ax$; the associated derivative with respect to $x$ is given by

$$\frac{d}{ dx } \; f(ax) = \frac{d \;f(ax) }{ d(ax)} \frac{d \; (ax)}{dx} = a f'(ax).$$

and

$$ \frac{d}{dx} K_{\nu} (x) = -\frac{1}{2} \left( K_{\nu-1}(x) + K_{\nu+1}(x) \right).$$

Now, apply the product rule to your expression and make use of the above hints.