The Derivative of a Function

One of the first topics studied in elementary calculus is the derivative of a real-valued function $f$. We will now formally define the derivative of a function below and begin to look at some of the properties of derivatives.

Definition: Let $f$ be a function defined on the open interval $(a, b)$ and let $c \in (a, b)$. Then $f$ is said to be Differentiable at $c$ if $\displaystyle{f'(c) = \lim_{x \to c} \frac{f(x) - f(x)}{x - c}}$ exists, where $f'(c)$ is called the Derivative of $f$ at $c$. The real-valued function $f'$ on $(a, b)$ for which $f'(c)$ exists is called the Derivative of $f$. The process by which $f'$ is obtained from $f$ is called Differentiation.

There are many notations for the derivative function including

(1)

\begin{align} \quad f'(x) \quad , \quad \frac{dy}{dx} \quad , \quad Df(x) \end{align}

We will commonly use the notations "$f'(x)$" and "$\frac{dy}{dx}$".

We will now look at a nice theorem which gives us an alternative definition for a function $f$ to be differentiable at a point $c \in (a, b)$ which is sometimes more convenient to use.

Theorem 1: Let $f$ be a function defined on the open interval $(a, b)$ and let $c \in (a, b)$. Then $f$ is differentiable at $c$ if and only if $\displaystyle{\lim_{h \to 0} \frac{f(c + h) - f(c)}{h}}$ exists.

• Proof: Let $f$ be differentiable at $c$. Then:

(2)

\begin{align} \quad f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \end{align}

• Let $h = x - c$. Then as $x \to c$, $h \to 0$, notice that:

(3)

\begin{align} \quad f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \end{align}

• $\Rightarrow$ If $f$ is differentiable, $\displaystyle{\lim_{h \to 0} \frac{f(c + h) - f(c)}{h}}$ must exist.

• $\Leftarrow$ Conversely, if $\displaystyle{\lim_{h \to 0} \frac{f(c + h) - f(c)}{h}}$ exists then $f'(c)$ exists. $\blacksquare$

Example 1

Apply Theorem 1 to show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2 - 2x$ is differentiable at any $c \in \mathbb{R}$ and compute $f'(3)$.

Using Theorem 1 we have that for any $c \in \mathbb{R}$:

(4)

\begin{align} \quad f'(c) &= \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \\ \quad &= \lim_{h \to 0} \frac{[(c + h)^2 - 2(c + h)] - [c^2 - 2c]}{h} \\ \quad &= \lim_{h \to 0} \frac{c^2 + 2ch + h^2 -2c - 2h - c^2 + 2c}{h} \\ \quad &= \lim_{h \to 0} \frac{2ch + h^2 - 2h}{h} \\ \quad &= \lim_{h \to 0} [2c + h - 2] \\ \quad &= 2c - 2 \end{align}

Plugging in $c = 3$ gives us that:

(5)

\begin{align} \quad f'(3) = 2(3) - 2 = 4 \end{align}