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