Differential Quotient

analysis

Definition

Differential Quotient

The differential quotient (or derivative) of a function $f$ at a point $x_0$ is the limit of the difference quotient as the interval $\Delta x$ approaches zero.

The difference quotient represents the slope of the secant line through $(x_0,f(x_0))$ and $(x_1,f(x_1))$:

$$\frac{\Delta f}{\Delta x}=\frac{f(x_1)-f(x_0)}{x_1-x_0}$$

The differential quotient is the slope of the tangent line at $x_0$:

$$\frac{df}{dx}=\underset{\Delta x\to 0}{\lim }\frac{\Delta f}{\Delta x}=\underset{x_1\to x_0}{\lim }\frac{f(x_1)-f(x_0)}{x_1-x_0}$$

Significance

The differential quotient provides a formal way to calculate the instantaneous rate of change of a function. If this limit exists at $x_0$, the function is said to be differentiable at that point.