The Derivative

The derivative of the graph of a smooth function at a point is the slope of the line tangent to this graph at that point. More formally, let f:D->R be a real function defined on the nonempty, open domain D, and let x belong to D. We say f is differentiable at x iff the following limit exists at x: lim (f(y)-f(x))/(y-x). This limit is f's derivative at x: f'(x). If f is differentiable at every point of D, the function g:D->R such that g(x)=f'(x) is called f's derivative.