Derivative of a function of one variable

The fundamental definition of the Differential Calculus is:

Definition 4.4.1   The derivative^4.4 of a function is the limit of the ratio of the increment of the function to the increment of the independent variable, when the latter increment varies and approaches the limit zero.

When the limit of this ratio exists, the function is said to be differentiable, or to possess a derivative.

The above definition may be given in a more compact form symbolically as follows: Given the function

$$y = f(x),$$ (4.2)

and consider $x$ to have a fixed value.

Let $x$ take on an increment $\Delta\, x$; then the function $y$ takes on an increment $\Delta\, y$, the new value of the function being

$$y + \Delta\, y = f(x + \Delta\, x).$$ (4.3)

To find the increment of the function, subtract (4.2) from (4.3), giving

$$\Delta\, y = f(x + \Delta\, x) - f(x).$$

Dividing by the increment of the variable, $\Delta\, x$, we get

$$\frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}.$$ (4.4)

The limit of this ratio when $\Delta\, x$ approaches the limit zero is, from our definition, the derivative and is denoted by the symbol $\frac{dy}{dx}$. Therefore

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}.$$

defines the derivative of $y$ [or $f(x)$] with respect to $x$. From (4.3), we also get

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

The process of finding the derivative of a function is called differentiation.

It should be carefully noted that the derivative is the limit of the ratio, not the ratio of the limits. The latter ratio would assume the form $\frac {0}{0}$, which is indeterminate (§3.2).