Differentiation

Abstract

Let $\mathbb{R}^{\rm N} \to \mathbb{R}$. By the partial derivative of f with respect to its i th variable we mean the function

$$ Dif(x) = \mathop {\lim }\limits_{\lambda \to 0} \frac{{f(x + \lambda ei) - f(x)}} {\lambda } $$

Remember that e_i is the vector with 1 in the ith coordinate and 0 everywhere else.This is also denoted by the symbol \frac{{\partial (x)}} {{\partial x_i }} The domain of this function is, of course, the set of all x for which the limit exists. We recall from calculus that in terms of Computing a partial derivative from a given function, we simply regard all variables except the ith one as constants and apply standard differentiation rules.