I have a standard normal distribution. cdf stands for cumulative distr. function and pdf for probability distribution function.

since the cdf is an intergral of pdf, the pdf is the derivative of cdf.

I have the function of the form:

f(x) = cdf(x)-cdf(-x)

I want a 1st derivative of this function, but if I just use

f'(x) = pdf(x) - pdf(-x) it is always zero, since the distribution is symmetrical.

This makes no sense to me, the derivative should be zero for x where pdf(x)~0, but not zero near zero.

Where's my mistake?

Thanks in advance.