Chapter 3 Differentiability
As with continuity, we offer three equivalent definitions of derivative, one geometric, topological, and one analytical.
Definition 3.1
The non-vertical line \(L\) is tangent to the function \(f\) at the point \(P=(x,y)\) means that:
\(x\) is a limit point of the domain of \(f\text{,}\)
\(P\) is a point of \(L\text{,}\) and
if \(A\) and \(B\) are non-vertical lines containing \(P\) with the line \(L\) between them (except at \(P\)), then there are two vertical lines \(H\) and \(K\) with \(P\) between them such that if \(Q\) is a point of \(f\) between \(H\) and \(K\) which is not \(P\text{,}\) then \(Q\) is between \(A\) and \(B\text{.}\)
In the previous definition we write that we have three distinct lines, \(A, B,\) and \(L\) with \(L\) between \(A\) and \(B\) (except at \(P\)). By this we mean that for any point \(l\) on \(L\) (except \(P\)) there is a point \(a\) on \(A\) and a point \(b\) on \(B\) so that either \(a\) is below \(l\) which is below \(b\) or that \(b\) is below \(l\) which is below \(a\text{.}\)
Definition 3.2
If \(f\) is a function, then the statement that \(f\) has a derivative at the number \(a\) in the domain of \(f\) means that \(f\) has a non-vertical tangent line at the point \((a,f(a))\text{.}\) The slope of the line tangent to \(f\) at the point \((a,f(a))\) is called the derivative of \(f\) at \(a\).
Definition 3.3
If \(f\) is a function, the statement that \(f\) has derivative \(D\) at the number \(x\) in the domain of \(f\) means that
\(x\) is a limit point of the domain of \(f\text{,}\) and
-
if \(S\) is an open interval containing \(D\text{,}\) then there is an open interval \(T\) containing \(x\) such that if \(t\) is a number in \(T\) and in the domain of \(f\) and \(t \ne x\text{,}\) then
\begin{equation*}
{f(t)-f(x) \over t-x} \in S.
\end{equation*}
As an alternative to this definition:
Definition 3.4
If \(f\) is a function, the statement that \(f\) has derivative \(D\) at the number \(x\) in the domain of \(f\) means that
\(x\) is a limit point of the domain of \(f\text{,}\) and
if \(\epsilon\) is a positive number, then there is a positive number \(\delta\) such that if \(t\) is in the domain of \(f\) and \(|t-x| \lt \delta\) then \(\displaystyle \left| {f(t)-f(x) \over t-x} - D \right| \lt \epsilon\text{.}\)
Problem 3.5
Use any of the definitions of derivative to show that if \(f\) is the function defined by the expression \(f(x)=x^2 + 1\) for all numbers \(x\) then \(f\) has derivative \(6\) at \(3\text{.}\)
Problem 3.6
Use the definition of tangent to show that if \(f\) is a function whose domain includes \((-1,1),\) and for each number \(x\) in \((-1,1)\text{,}\) \({-x^2 \le f(x) \le x^2}\text{,}\) then the \(x\)-axis is tangent to \(f\) at the point \((0,0)\text{.}\)
Problem 3.7
Use any of the definitions of derivative except Definition 3.2 to show that if \(f\) is a function whose domain includes \((-1,1)\) and for each number \(x\) in \((-1,1)\text{,}\) \({-x^2 \le f(x) \le x^2}\text{,}\) then the derivative of \(f\) at the point \((0,0)\) is \(0\text{.}\)
Theorem 3.8
If \(f\) is a function, and \(x\) is in the domain of \(f\text{,}\) then \(f\) does not have two tangent lines at the point \((x,f(x))\text{.}\)
Definition 3.9
If \(f\) is a function which has a derivative at some point, then the derivative of \(f\) is the function denoted by \(f'\text{,}\) such that for each number \(x\) at which \(f\) has a derivative, \(f'(x)\) is the derivative of \(f\) at \(x\text{.}\)
Problem 3.10
If \(M\) is a point set and \(p\) is a limit point of the set of limit points of \(M\text{,}\) then \(p\) is a limit point of \(M\text{.}\)
Definition 3.11
If \(M\) is a point set, then the closure of \(M\) is the set consisting of \(M\) together with any limit points of \(M\text{.}\) It is denoted by \(Cl(M)\) or by \(\overline M\text{.}\)
Theorem 3.12
If \(M\) is a point set then \(Cl(M)\) is a closed point set.
From this point forward we may use \(\mathbb{R}\) to represent the set of real numbers and \(D_f\) to denote the domain of \(f\text{.}\)
Theorem 3.13
Suppose that \(f\) is a function that is differentiable at the point \(p\) and that \(c \in \mathbb{R}\text{.}\) Show that the function \(g\) defined by \(g(x) = cf(x)\) for all \(x \in D_f\) is also differentiable at the point \(p\text{.}\)
Theorem 3.14
Suppose that each of \(f\) and \(g\) are functions that are differentiable at the point \(p\) and that \(h\) is the function defined by \(h(x) = f(x) + g(x)\) for all \(x \in D_f\text{.}\) Show that \(h\) is also differentiable at the point \(p\text{.}\)
Theorem 3.15
If \(f\) is a function, \(x\) is in the domain of \(f\text{,}\) and \(f\) has a derivative at \((x,f(x))\text{,}\) then \(f\) is continuous at \((x,f(x))\text{.}\)
Theorem 3.16
If \(f\) is a function having domain \([a,b]\text{,}\) \(x \in (a,b)\text{,}\) \(f(x) \ge f(t)\) for all \(t \in (a,b)\text{,}\) and \(f\) has a derivative at \(x\text{,}\) then \(f'(x)=0\text{.}\)
Problem 3.17
Does there exist a function \(f\) defined and continuous on \([0,1]\) such that \(f(0)=0\) and \(f(1)=1\) and \(f'(x)=0\) at all but countably many points of \([0,1]\text{?}\)
We can't prove everything we need, but at this point, you could prove that all polynomials are differentiable. You could also prove all the theorems about differentiability: the power rule, constant rule, sum rule, product rule, and quotient rules.