# Derivative of the error function

I got stuck with the derivative of the following function: $$erf(\frac{logit(\theta)-\mu}{\sqrt {2\sigma^2}})$$

with respect to $$\theta$$.

Are there handy approximations with elementary functions in that case?

Any help will be appreciated, thanks in advance!

The error function is defined by $$\text{erf}(x)=\frac{1}{\sqrt{\pi}}\int_0^{x}e^{-t^2}\mathrm{d}t$$. Therefore $$\frac{\mathrm{d}}{\mathrm{d}x}\text{erf}(x)=\frac{2}{\sqrt{\pi}}e^{-x^2}$$.
Set $$x=\frac{\text{logit}(\theta)-\mu}{\sqrt{2\sigma^2}}$$ and use the chain rule. Remember that $$\text{logit}\theta$$ is defined as $$\frac{\theta}{1-\theta}$$ which differentiates to $$\frac{\mathrm{d}}{\mathrm{d}\theta}\text{logit}\theta=\frac{-1}{(1-\theta)^2}$$.