Section 3.2 The Derivative as a Function
The formula for the tangent slope at one point
\(x=a\) on curve
\(y=f(x)\) can also be seen as a function, with argument
\(x\) and value the slope at the corresponding point:
Definition 3.2.1. Derivative.
The function \(f'\) given by
\begin{equation*}
f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}
\end{equation*}
is the derivative of function \(f\text{.}\) The domain of \(f'\) is the \(x\) values for which this limit exists; this can be smaller than the domain of \(f\text{.}\)
(The name comes from the ideas that this new function is
derived from the original one.)
One basic physical example is that when \(f(t)\) is position as a function of time, \(f'(t)\) is velocity as a function of time. We will study the geometry of this new function, and some useful things that it can tell us about the original function.
This helps with questions like getting information about position from measurements of velocity.
Example 3.2.2.
Example 3.2.3.
If \(f(x)=x^3-x\text{,}\) find a formula for \(f'(x)\text{.}\)
Illustrate by comparing the graphs of \(f\) and \(f'\text{.}\)
Example 3.2.4.
For \(f(x)=\sqrt{x}\text{,}\) find the derivative \(f'\text{.}\)
What is the domain of \(f'\text{?}\)
Example 3.2.5.
Calculate the derivative of \(f(x) = (1-x)/(2+x)\text{.}\)
Other Notations.
Many different notations are used for the derivative of \(y=f(x)\text{.}\) Besides \(f'\) as above (introduced by Joseph-Louis Lagrange) we often use \(\frac{dy}{dx}\) or \(\frac{df}{dx}\text{,}\) introduced by Gottfried Leibniz. Another notation is \(Df\text{,}\) due to Leonhard Euler: it will be used less here, but can be convenient at times; when it is important to identify the independent variable, the variant \(D_x f\) is used. So here is a collection of synonyms:
\begin{equation*}
f'(x) = y' = \frac{dy}{dx} = \frac{df}{dx} = \frac{d}{dx}f = Df(x) = D_x f(x).
\end{equation*}
Motivation for the Leibniz Notation.
The Leibniz notation is suggested by another way to write the formula for the derivative:
write \(\Delta x\) for \(h\text{,}\) the change in argument \(x\text{,}\) and let \(\Delta y = f(x+h)-f(x)\) be the corresponding change in \(y\text{.}\) Then
\begin{equation*}
\frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}
\end{equation*}
and the change in notation from a capital \(\Delta\) to a small \(d\) suggests the very small changes involved. In fact, Leibniz and other pioneers of calculus thought in terms of the derivative as the ratio of “infinitely small” (“infinitesimally small”) increments in both the \(x\) and \(y\) values: the precise use of the limit idea came later. This connection back to the division used to define the derivative is useful in some calculations later.
Differentiability: at Some Values, and Everywhere.
Limits do not always exist, so the derivative does not always exist. Thus, similar to continuity we have:
Definition 3.2.6.
A function \(f\) is differentiable at \(\mathbf{a}\) if \(f'(a)\) exists.
It is differentiable on an open interval \((a,b)\text{,}\) \((-\infty,a)\text{,}\) \((a,\infty)\) or \((-\infty,\infty)\) if it is differentiable at every number in that interval.
If a function is differentiable at every number in its domain, we simply call it differentiable.
Example 3.2.7.
Differentiability and Continuity.
The above example shows that sometimes, a function can be continuous at \(a\text{,}\) but not differentiable at \(a\text{.}\) The opposite is not true however:
Theorem 3.2.8.
If a function is differentiable at \(a\text{,}\) it is also continuous at \(a\text{.}\)
Proof.
Differentiability says that the limit \(f'(a)=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\) exists. Using the product law for limits,
\begin{align*}
\lim_{h \to 0}f(a+h)-f(a) \amp=\amp \lim_{h \to 0}\left(h \cdot \frac{f(a+h)-f(a)}{h} \right)\\
\amp=\amp \left(\lim_{h \to 0} h\right) \left(\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\right)\\
\amp=\amp 0 \cdot f'(a) = 0.
\end{align*}
Then the addition rule for limits gives
\begin{align*}
\lim_{h \to 0}f(a+h) \amp=\amp \lim_{h \to 0}[(f(a+h)-f(a))+f(a)]\\
\amp=\amp \lim_{h \to 0}[f(a+h)-f(a)] + \lim_{h \to 0} f(a) = 0 + f(a) = f(a).
\end{align*}
With \(x = a+h\text{,}\) as \(x \to a\text{,}\) \(h = x-a \to 0\text{,}\) so
\begin{equation*}
\lim_{x \to a} f(x) = \lim_{h \to 0} f(a+h) = f(a): \text{ continuity at } a.
\end{equation*}
How Can a Function Fail to be Differentiable?
The above theorem tells us one way that a function can fail to be differentiable at \(a\text{:}\) if \(f\) is not continuous at \(a\text{,}\) it is not differentiable there either. So a function is non-differentiable at jump discontinuities, removable discontinuities, places where it has vertical asymptotes, and places where wilder behavior occurs, as with \(\sin(1/x)\) at 0. But Example 5 shows another situation, where a function can fail to be differentiable at a point even though it is continuous there. The problem there is a “corner” at the origin, where the graph does not have a well defined tangent line. In fact any line through the origin of slope between \(-1\) and \(1\) is “tangent” to \(y=|x|\) at the origin, in that it touches the curve but does not cross it. One other situation where the derivative does not exist is when a graph effectively has a vertical tangent line, or an infinite slope at a point.
Example 3.2.9. No derivative at a point due to a vertical tangent.
Consider \(y=f(x) = x^{1/3}\text{.}\) It is continuous everywhere, but if we try to compute the derivative at \(x=0\text{,}\) we get
\begin{equation*}
\lim_{h \to 0}\frac{h^{1/3}-0^{1/3}}{h} = \lim_{h \to 0} \frac{h^{1/3}}{h} = \lim_{h \to 0} \frac{1}{h^{2/3}}, \text{ DNE.}
\end{equation*}
Actually there is an infinite limit, so in a sense the graph has an infinite tangent slope at the origin: a vertical tangent line.
Continuity vs Differentiability.
Intuitively,
the graph of a continuous function has no breaks, whereas
the graph of a differentiable function has no breaks, sharp corners, or vertical tangents.
Second Derivatives.
If function \(f\) is differentiable, its derivative \(f'\) is also a function, and so can have a derivative, called the second derivative of \(f\text{.}\) The Lagrange notation adds another prime: \((f')'\) or more commonly \(f''\text{;}\) The Leibnitz notations adds another “\(d/dx\)”; the Euler notation adds another “factor” of D:
\begin{equation*}
f' = \frac{d}{dx}f = \frac{df}{dx} = Df, \; (f')' = f'' = \frac{d}{dx}\frac{d}{dx}f = \frac{d}{dx}\frac{df}{dx} = \frac{d^2 f}{dx^2} = D^2 f.
\end{equation*}
The more compact version of Leibnitz notation treats each “\(d\)” on the top and each “\(dx\)” on the bottom as if they were factors in a fraction.
Example 3.2.10.
For \(f(x)=x^3-x\text{:}\)
Third and Higher Derivatives.
The above can be repeated, leading to the third derivative of \(f\text{,}\) denoted \(f'''\) or \(\frac{d^3f}{dx^3}\) or \(D^3 f\text{,}\) and so on to the \(n\)-th derivative for any natural number \(n\text{.}\) The primes can get messy when we do this too often, so the \(n\)-th derivative of \(f\) (it it exists) is also denoted \(f^{(n)}\text{,}\) or \(\frac{d^n f}{dx^n}\text{.}\) The Euler notation is perhaps the most elegant: \(D^n f\text{.}\) Note the parentheses in the Lagrange form: \(f^{(n)}(x)\) is different from \(f^n(x)\text{,}\) which is \([f(x)]^n\text{.}\)
Example 3.2.11.
For \(f(x)=x^3-x\) as above:
Calculate \(f^{(3)}\text{.}\)
Calculate \(f^{(4)}\text{.}\)
Calculate \(f^{(17)}\text{.}\)
Exercises Exercises
Study
Calculus Volume 1, Section 3.2 2 ; in particular all Examples and Checkpoint items are worth reviewing, along with Exercises 55, 57, 65, 67, 79, 80 and 96.
openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function
openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function
