Differentiability in One Variable

Recall that a function f of one variable is differentiable at a point x₀if the limit of the difference quotient

$$\lim _{x\rightarrow {x_{0}}}\,\frac {f(x) - f(x_{0})}{x - x_{0}}$$

exists. If it exists, the value which is the limit is called the derivative of f at x₀. The derivative is the slope of the tangent line to the graph at x₀ and the equation of y = f(x₀) + f`(x₀)(x-x₀). Intuitively, we can check to see if f is differentiable at x₀ by ``zooming-in'' on the point $({x_{0}}, \,{f}({x_{0}}))$ on the graph of f. If, as we zoom in on the point, the graph of the function appears to be the graph of a straight line, then f is differentiable at x<sub>0</sub>. Thus for functions of one variable, the graph of f looks like a straight line as we ``zoom in'' near x₀ if and only if the derivative of f exists at x₀.