Differentiability

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 the point \(x=x_0\text{.}\) Geometrically, it is the slope of the tangent at the point \((x_0,f(x_0))\) of the graph \(y=f(x)\text{.}\)

Given a description of how a quantity \(y\) changes with respect to a quantity \(x\) as a function \(y(x)\text{,}\) we often interested in how fast \(y\) is changing against \(x\text{.}\) The rate of change of \(y\) with respect to the change of \(x\) is easy to understand: it is simply the ratio of the change of \(y\) over the change of \(x\text{.}\) That is,

\begin{equation*} \frac{\Delta y}{\Delta x} = \frac{y(x_2)-y(x_1)}{x_2-x_1}. \end{equation*}

For instance, if you traveled 140 miles in 2 hours then your average speed over that 2 hours period is \(140/2 = 70\) miles per hour (mph). But, as you can imagine, an average speed of \(70\) mph, does not mean you traveled constantly as speed \(70\) mph during that 2 hour period.

Things indeed get trickier when we are interested in the instantaneous rate of change of \(y\) with respect to \(x\text{.}\) In view of the example above, that is, we are interested in the speedometer reading of your car. Suppose the quantity \(y\) is given by a function \(f\) of \(x\) and we are interested in the instantaneous rate of change of \(y\) with respect to \(x\) when \(x\) is at the value \(x_0\text{.}\) As we have discussed, for \(x \neq x_0\)

\begin{equation*} \varphi_{x_0}(x) = \frac{f(x)-f(x_0)}{x-x_0} \end{equation*}

is the rate of change of \(y\) with respect to \(x\text{.}\) However, we cannot simply evaluate \(\varphi_{x_0}(x)\) at \(x=x_0\) to obtain the instantaneous rate of change as the denominator vanishes when \(x =x_0\text{.}\) And yet, we do expect \(\varphi_{x_0}(x)\) is close to the instantaneous rate of change that we are looking for when \(x\) is close to \(x_0\text{.}\) Hence, it 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 \(\varphi_{x_0}(x)\) continuous at \(x=x_0\) such that for all \(x\) at which \(f\) is defined,

\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 \(x=x_0\text{.}\)

Remark 1.2.

This definition of derivative is due to Constantin Caratheodory.
We are simply defining differentiability in terms of continuity. But we hope continuity is a more intuitive concept and is easier to understand.
For the expert: it is implicit in the definition that \(f\) is defined at \(x_0\text{.}\) Moreover, the uniqueness of the extension of \(\varphi_{x_0}(x)\) is equivalent to \(x_0\) being a limit point of the domain of \(f\text{.}\) Note also that the uniqueness of the extension is automatic when the domain of \(f\) is an interval (with at least two distinct points).
Note also that differentiability is a local property. That means if \(f\) and \(g\) agree on an open interval containing \(a\text{,}\) then \(f'(a)=g'(a)\) if either of them exists.

Geometrically \(\varphi_{x_0}(x)\) is the slope of the secant line joining the point \((x_0,f(x_0))\) and \((x,f(x))\text{.}\) So, if \(f(x)\) is differentiable at \(x=x_0\text{,}\) then as \(x\) approaches \(x_0\) these secants approach the line defined by

\begin{equation} f(x_0) + f'(x_0)(x-x_0) \tag{1.2} \end{equation}

which is called the tangent of \(f(x)\) at \(x=x_0\text{.}\) It is the line that passes through the point \((x_0, f(x_0))\) with slope \(f'(x_0)\text{.}\) In other words \(f'(x_0)\text{,}\) if exists, is the slope of the tangent of the graph \(y=f(x)\) at the point \((x_0,f(x_0))\text{.}\)

Using \(f'(x_0)\) to denote the derivative of \(f(x)\) at \(x =x_0\) is due to Lagrange. This notation suggests that the derivatives of \(f(x)\) at various points are the values of another function namely \(f'(x)\text{.}\) Another popular notation for the derivative of \(f(x)\) at \(x = x_0\) is \(\ds \frac{df}{dx}(x_0)\text{.}\) This notation is due to Leibniz. It reminds us that the derivative is the rate of change of \(f(x)\) with respect to \(x\) at \(x_0\text{.}\) It emphasizes the variable is \(x\text{.}\) If we do not a specific name of the function but simply consider its values as a dependent valuable \(y\) depending on \(x\text{,}\) then we also write its 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 1.3.

For a linear function \(f(x)=mx +b\text{,}\) it is plain 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 \(f(x)=mx+b\) (as a function of \(x\)) 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 1.4.

Consider the function \(f(x)=1/x\text{.}\) For \(x, x_0 \neq 0\text{,}\) since

\begin{equation*} f(x)-f(x_0)=\frac{1}{x} - \frac{1}{x_0} = -\frac{1}{xx_0}(x-x_0) \end{equation*}

and the function \(\varphi_{x_0}(x) = -1/(xx_0)\) is continuous at \(x_0\text{,}\) we conclude that \(f'(x_0) = -1/x_0^2\text{.}\) And so the derivative of \(1/x\) is the function \(-1/x^2\text{.}\)

Proposition 1.5.

If \(f\) is differentiable at \(x_0\) then \(f\) is continuous at \(x_0\text{.}\) In particular, differentiable functions are continuous.

Proof.

By definition of differentiability,

\begin{equation*} f(x) = f(x_0) + \varphi(x)(x-x_0) \end{equation*}

for some function \(\varphi(x)\) that is continuous at \(x=x_0\text{.}\) Therefore, \(f\) is continuous at \(x_0\) since constant functions are continuous and continuous functions are closed under sum and products.

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