1.7.1—Gradient as limit

The derivative f′(x) is lim_{h→0}[f(x+h)−f(x)]/h when the limit exists. It is the gradient of the tangent and measures instantaneous rate of change.

A secant gradient uses two distinct points; the tangent is its limiting value. At a sharp corner or cusp the two-sided derivative may not exist.

For f(x)=x², the difference quotient is 2x+h, so letting h→0 gives f′(x)=2x.

Substituting h=0 into the quotient before simplifying causes division by zero and is not differentiation.