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Section 3.2 The Derivative as a Function

Definition 3.2.1.

The derivative of the function \(f\) with respect to \(x\) is the function \(f'\) whose value at \(x\) is

\begin{equation*} f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} \end{equation*}

provided the limit exists.

Note that the domain of \(f'\) is the set of points where the limit does exist.

If \(f'\) exists at a value \(x_0\text{,}\) we say that \(f\) is differentiable at \(x_0\text{;}\) if \(f'\) exists at every point in the domain of \(f\text{,}\) then we say that \(f\) is differentiable.

An alternate definition: \(\displaystyle\lim_{x\rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}\)

Note on language: differentiation is the process of finding the derivative; to differentiate is the verb for computing the derivative.

Example 3.2.2.

Find the derivatives of:

  1. \(\displaystyle f(x)=\sqrt{x}\)

  2. \(\displaystyle g(x)=x^2+5x-7\)

Solution.

  1. \begin{align*} f'(x) \amp =\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\\ \amp = \lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}\\ \amp = \lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\\ \amp = \lim_{h\to 0} \frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})}\\ \amp = \lim_{h\to 0} \frac{h}{h(\sqrt{x+h}+\sqrt{x})}\\ \amp = \lim_{h\to 0} \frac{1}{\sqrt{x+h}+\sqrt{x}}\\ \amp = \frac{1}{\sqrt{x+0}+\sqrt{x}}\\ \amp = \frac{1}{2\sqrt{x}} \end{align*}

  2. \begin{align*} g'(x) \amp =\lim_{h\to 0} \frac{g(x+h)-g(x)}{h}\\ \amp = \lim_{h\to 0} \frac{((x+h)^2+5(x+h)-7) - (x^2+5x-7)}{h}\\ \amp = \lim_{h\to 0} \frac{(x^2+2xh +h^2+5x+5h-7) - (x^2+5x-7)}{h}\\ \amp = \lim_{h\to 0} \frac{2xh +h^2+5h}{h}\\ \amp = \lim_{h\to 0} \frac{h(2x +h+5)}{h}\\ \amp = \lim_{h\to 0} \frac{2x +h+5}{1}\\ \amp = 2x+0+5\\ \amp = 2x+5 \end{align*}

Some notation: Given \(y=f(x)\text{:}\)

\begin{equation*} f'(x)=\frac{df}{dx}=\frac{d}{dx}f(x)=D(f)(x)=D_xf(x)=y'=\frac{dy}{dx} \end{equation*}

For the value of the derivative at a specific point:

\begin{equation*} f'(9) = \left.\frac{df}{dx}\right|_{x=9}=\left.\frac{dy}{dx}\right|_{x=9}=\left.\frac{d}{dx}f(x)\right|_{x=9}=D_xf(9) \end{equation*}

Example 3.2.3.

The graphs of \(y=f(x)\) and \(y=g(x)\) are shown. Sketch the graphs of the derivatives.

Graph of parabola
Figure 3.2.4. \(y=f(x)\)
Graph of piecewise function
Figure 3.2.5. \(y=g(x)\)

Definition 3.2.6.

\(f\) is differentiable on an open interval if the derivative exists at every point in the interval.

\(f\) is differentiable on a closed interval \([a,b]\) if it is differentiable on \((a,b)\) and if both

  • \(\displaystyle\lim_{h\rightarrow 0^+} \frac{f(a+h)-f(a)}{h}\) (right-hand derivative at \(a\)), and

  • \(\displaystyle\lim_{h\rightarrow 0^-} \frac{f(b+h)-f(b)}{h}\) (left-hand derivative at \(b\))

exist.

Note that the derivative exists at a point only if the left- and right-hand derivatives both exist and agree.

Example 3.2.7.

Let \(\displaystyle f(x) = \begin{cases} \frac{3}{x}\amp x\leq 3\\ x-2 \amp x \gt 3 \end{cases}\text{.}\) Is \(f\) differentiable at \(x=3\text{?}\)

Graph of piecewise function
Figure 3.2.8. \(y=f(x)\)

Examples of how a function can be not differentiable.

Corner

As in the example above.

Vertical tangent

For example, \(y=\sqrt[3]{x}\)

Graph of function with vertical tangent
Figure 3.2.9. \(y=\sqrt[3]{x}\)
Cusp

The one-sided derivatives are \(\pm\infty\text{,}\) with opposite signs, e.g., \(\displaystyle f(x) = \sqrt{|x|} = \begin{cases} \sqrt{-x} \amp x \lt 0\\ \sqrt{x} \amp x \geq 0 \end{cases}\)

Graph of function with cusp
Figure 3.2.10. \(y=\sqrt{|x|}\)
Discontinuity

A function is non-differentiable at any point of discontinuity.

Note that the converse is false: a function can be continuous and not differentiable.