Skip to main content

Calculus An intuitive approach

Wai Yan Pong

Section 1.1 Differentiability

The derivative of a function \(f(x)\) at a point \(x_0\) is the instantaneous rate of change of \(f(x)\) with respect to \(x\) at \(x=x_0\text{.}\) Geometrically, it is the slope of the tangent line to the graph \(y=f(x)\) at the point \((x_0, f(x_0))\text{.}\)
🔗
Given a description of how a quantity \(y\) changes with respect to a quantity \(x\) via a function \(y=f(x)\text{,}\) we are often interested in how fast \(y\) is changing relative to \(x\text{.}\) The rate of change of \(y\) with respect to \(x\) is straightforward to understand: it is simply the ratio of the change in \(y\) to the change in \(x\text{.}\) That is,
\begin{equation*} \frac{\Delta y}{\Delta x} = \frac{f(x_2)-f(x_1)}{x_2-x_1}. \end{equation*}
For instance, if you traveled 140 miles in 2 hours, then your average speed over that two-hour period is \(140/2 = 70\) miles per hour (mph). However, an average speed of 70 mph does not mean you traveled at a constant speed of 70 mph throughout that entire period.
🔗
Things get trickier when we are interested in the instantaneous rate of change of \(y\) with respect to \(x\text{.}\) This is analogous to the speedometer reading in your car at a specific moment. Suppose the quantity \(y\) is given by a function \(f\) of \(x\text{,}\) and we want to find the instantaneous rate of change when \(x\) is at the value \(x_0\text{.}\) As we have discussed, for \(x \neq x_0\text{,}\)
\begin{equation*} \varphi_{x_0}(x) = \frac{f(x)-f(x_0)}{x-x_0} \end{equation*}
is the (average) rate of change of \(y\) with respect to \(x\) over the interval between \(x\) and \(x_0\text{.}\) We cannot simply evaluate \(\varphi_{x_0}(x)\) at \(x=x_0\) because the denominator vanishes. Nevertheless, we expect \(\varphi_{x_0}(x)\) to be close to the instantaneous rate of change when \(x\) is close to \(x_0\text{.}\) Thus, it is reasonable to say that if \(\varphi_{x_0}(x)\) can be extended to a function that is continuous at \(x=x_0\text{,}\) then the value of this extension at \(x=x_0\) is the instantaneous rate of change of \(y\) at \(x=x_0\text{.}\)
🔗

Definition 1.1.

A function \(f(x)\) is differentiable at \(x=x_0\) if there exists a unique function \(\varphi_{x_0}(x)\text{,}\) continuous at \(x=x_0\), such that for all \(x\) in the domain of \(f\text{,}\)
\begin{equation} f(x)-f(x_0) = \varphi_{x_0}(x)(x-x_0).\tag{1.1} \end{equation}
The value \(\varphi_{x_0}(x_0)\text{,}\) also denoted by \(f'(x_0)\text{,}\) is called the derivative of \(f(x)\) with respect to \(x\) at \(x=x_0\text{.}\)
🔗
🔗
Geometrically, \(\varphi_{x_0}(x)\) is the slope of the secant line joining the points \((x_0, f(x_0))\) and \((x, f(x))\text{.}\) If \(f(x)\) is differentiable at \(x=x_0\text{,}\) then as \(x\) approaches \(x_0\text{,}\) these secant lines approach the line defined by
\begin{equation*} y = f(x_0) + f'(x_0)(x-x_0) \end{equation*}
which is called the tangent line to \(f(x)\) at \(x=x_0\text{.}\) This is the line passing through the point \((x_0, f(x_0))\) with slope \(f'(x_0)\text{.}\) In other words, \(f'(x_0)\text{,}\) if it exists, is the slope of the tangent to the graph \(y=f(x)\) at the point \((x_0, f(x_0))\text{.}\)
🔗
The notation \(f'(x_0)\) for the derivative of \(f(x)\) at \(x=x_0\) is attributed to Lagrange. This notation suggests that the derivatives of \(f(x)\) at various points are values of another function, namely \(f'(x)\text{.}\) Another popular notation for the derivative is \(\frac{df}{dx}(x_0)\text{,}\) due to Leibniz. This notation emphasizes that the derivative is the rate of change of \(f\) with respect to \(x\text{.}\) If we do not have a specific name for the function but simply consider its values as a dependent variable \(y\) depending on \(x\text{,}\) we also write the derivative at \(x=x_0\) as
\begin{equation*} \frac{dy}{dx}(x_0), \quad \left.\frac{dy}{dx}\right|_{x=x_0}, \quad \text{or} \quad \left.\frac{dy}{dx}\right|_{(x_0, f(x_0))}. \end{equation*}
🔗

Example 1.2.

For a linear function \(f(x)=mx +b\text{,}\) it is easy to verify that
\begin{equation*} f(x) - f(x_0) = m(x-x_0) \end{equation*}
for any \(x, x_0\text{.}\) Constant functions are continuous, so we have \(f'(x_0) = m\) for any \(x_0\text{.}\) In other words, the derivative of \(f(x)=mx+b\) is the constant function \(f'(x)=m\text{.}\) In particular, the derivative of a constant function \(f(x)=b\) is the zero function, and the derivative of the identity function \(f(x)=x\) is the constant function \(1\text{.}\)
🔗
🔗

Example 1.3.

Consider the function \(f(x)=1/x\text{.}\) For \(x, x_0 \neq 0\text{,}\)
\begin{equation*} f(x)-f(x_0) = \frac{1}{x} - \frac{1}{x_0} = \frac{x_0 - x}{xx_0} = -\frac{1}{xx_0}(x-x_0). \end{equation*}
Since the function \(\varphi_{x_0}(x) = -1/(xx_0)\) is continuous at \(x_0\text{,}\) we conclude that \(f'(x_0) = -1/x_0^2\text{.}\) Thus, the derivative of \(1/x\) is the function \(-1/x^2\text{.}\)
🔗
🔗

Proof.

By the definition of differentiability, there exists a function \(\varphi(x)\) continuous at \(x=x_0\) such that
\begin{equation*} f(x) = f(x_0) + \varphi(x)(x-x_0). \end{equation*}
Since constant functions are continuous and the set of continuous functions is closed under addition and multiplication, \(f\) must be continuous at \(x_0\text{.}\)
🔗
🔗

Remark 1.5.

  1. The definition of differentiability presented here is due to Constantin Carathéodory.
    🔗
    🔗
  2. This approach defines differentiability in terms of continuity, leveraging the intuition that continuity is often easier to grasp.
    🔗
    🔗
  3. For the expert: It is implicit in the definition that \(f\) is defined at \(x_0\text{.}\) Furthermore, the uniqueness of the extension \(\varphi_{x_0}(x)\) is equivalent to \(x_0\) being a limit point of the domain of \(f\text{.}\) Uniqueness is guaranteed when the domain of \(f\) is an interval.
    🔗
    🔗
  4. Differentiability is a local property. If two functions \(f\) and \(g\) agree on an open interval containing \(a\text{,}\) then \(f'(a)=g'(a)\) if either derivative exists.
    🔗
    🔗
🔗
🔗
🔗
PrevTopNext