Definition of the derivative
The derivative compresses the tangent problem and the velocity problem into one object: a limit of difference quotients.
Definition
The derivative of f at x is the limit of the difference quotient as h approaches 0. It measures the instantaneous rate of change of f at that exact input. If the limit exists, f is called differentiable at x, and f' denotes the resulting new function.
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}Three readings
The same number carries three meanings. Geometrically it is the slope of the tangent line to the graph at x. Physically it is the instantaneous velocity when f describes position. Generally it is the rate at which the output changes per unit change of the input.
\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}Worked example
Take f(x) = x^2 - 4x. The difference quotient is ((x+h)^2 - 4(x+h) - x^2 + 4x)/h, which simplifies to 2x - 4 + h. Letting h approach 0 gives f'(x) = 2x - 4. At x = 1 the slope is -2, so the curve is falling there.
f'(x) = \lim_{h \to 0} (2x - 4 + h) = 2x - 4Flashcard
Limit definition of the derivative
Check your understanding
Using the limit definition, what is the derivative of f(x) = x^3 at x = 2?