Differentiation

Definition of a Derivative

Consider a function $f(x)$ represented by the curve in Figure 1. The derivative of $f(x)$ at a point $x=x_0$, denoted by $f’(x_0)$, is the slope of the tangent line to the graph of $f(x)$ at the point $(x_0, f(x_0))$. A tangent line is the limit of the secant lines joining points $P=(x_0, f(x_0))$ and $Q$ on the graph of $f(x)$ as $Q$ approaches $P$.

Figure 1: A graph with secant and tangent lines.

Figure 2: Geometric definition of the derivative.

The slope of secant $PQ$ is rise divided by run, or the ratio $\frac{\Delta f}{\Delta x}$ as shown in Figure 2. As $Q$ gets closer to $P$, the distance $\Delta x$ goes to zero. Then, the derivate which is equivalent to the slope of tangent, can be expressed mathematically as:

$\begin{align} m &= f’(x_0) \\ &= \lim_{Q \rightarrow P} \frac{\Delta f}{\Delta x} \nonumber \\ &= \lim_{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x} \\ &= \lim_{\Delta x \rightarrow 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} \end{align}$

The last equation above is the algebraic definition of a derivative.

Common derivative properties

When you take the derivative of an odd function, you always get an even function and vice versa.
Differentiable implies continuous: If $f$ is differentiable at $x_0$, then $f$ is continuous at $x_0$.

Proof

A function is continuous if $\lim_{x \rightarrow x_0} f(x) - f(x_0) = 0$. We multiply and divide this by the same value:

$\begin{align} \lim_{x \rightarrow x_0} f(x) - f(x_0) &= \lim_{x \rightarrow x_0} \frac{f(x)- f(x_0)}{x - x_0} \left(x - x_0\right) \\ &= f’(x) \cdot 0 \\ &= 0 \end{align}$

Notations

In calculus, as in the English language, there are many ways to express the same thing. Here we mention two notations most commonly used in calculus: Leibniz’ and Newton’s notations. Newton and Leibniz both invented calculus independently, and there has been anonymity between them, in addition to controversy about who has first invented calculus.

We let $y = f(x)$, where $y$ is a variable representing the function $f$ at any given $x$. From the formula for the derivative, we represent “the change in $y$” as $\Delta y = \Delta f = f(x_0 + \Delta x) - f(x_0)$. On the other hand, the “change in $x$” is $\Delta x = x - x_0$.

Leibniz’ notation

$\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} \equiv \frac{dy}{dx}$

Using Leibniz’ notation, we might also represent the derivative as $\frac{df}{dx}$, $\frac{d}{dx}f$, $\frac{d}{dx}y$. Notice that Leibniz’ notation does not specify where the derivative is being evaluated (e.g. at $x_0$). However, it expresses the derivative as a ratio, which is more convenient than Newton’s notation in certain situations.

Newton’s notation

The advantage of Newton’s nation is that is compact representation of the derivative:

$\lim_{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x} \equiv f^\prime(x_0)$

Last updated on Feb 15, 2019