Derive the derivative from the limit of the difference quotient.
The derivative f'(x) is the instantaneous rate of change of f at x — the slope of the tangent line to the graph at that point. It is defined as the limit of the average rate of change (difference quotient) as the interval shrinks to zero.
A function is differentiable at x = a if this limit exists. Differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| is continuous but not differentiable at 0).
f'(x) = lim_{h→0} [(x+h)² − x²]/h = lim_{h→0} [x² + 2xh + h² − x²]/h = lim_{h→0} [2x + h] = 2x.Remember This!
The difference quotient [f(x+h)−f(x)]/h represents the slope of a secant line. As h→0, the secant line rotates to become the tangent line, and its slope becomes the derivative.
Using the definition of the derivative, what is f'(x) if f(x) = 3x + 5?