Differential Quotient

analysis

Definition

Differential Quotient

The slope of the tangent is a measure of change in the function at a given point

The slope of the secant is given by:

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

which is the average change of $f(x)$ over the interval $[x_0,x_1]$ and is called the difference quotient. In order to obtain the slope of the tangent, we now let $\Delta x$ approach to 0 and compute the limit:

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

If the limit exists, then this quantity is called differential quotient.