Derivatives and Differentials of Functions
3.1 Definition and Calculation of Derivatives
A point \(x_0\) is called an interior point of \(I\), if there exists \(\delta > 0\) such that \[ \forall x,\ |x - x_0| < \delta \Rightarrow x \in I. \]
Let \(x_0\) be an interior point of \(I\). A function \(f: I \to \mathbb{R}\) is said to be differentiable at \(x_0\) if there exists a linear function \(L: \mathbb{R} \to \mathbb{R}\) such that
\[ f(x_0 + h) = f(x_0) + L(h) + o(h), \quad h \to 0. \]
In this case, \(L\) is called the differential of \(f\) at \(x_0\), denoted by \(df(x_0)\).
For a one-variable function, the differential \(df(x_0)\) is a linear function, that is, a proportional function whose proportionality constant is denoted by \(f'(x_0)\). This \(f'(x_0)\) is called the derivative of \(f\) at \(x_0\).
\[ df(x_0)(h) = f'(x_0)h. \]
\[ \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0) - f'(x_0)h}{h} = \lim_{h \to 0} \frac{o(h)}{h} = 0. \]
The derivative is defined as:
\[ f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}. \]
If this limit exists, we say that \(f\) is differentiable at \(x_0\).
- Traditional notation: Let \(\Delta x = h\), \(\Delta y = f(x_0 + \Delta x) - f(x_0)\), then
\[ f'(x_0) = \lim_{h \to 0} \frac{\Delta y}{\Delta x} = \left. \frac{dy}{dx} \right|_{x_0}. \]
Lagrange notation: \(f'(x_0).\)
Leibniz notation: \(\displaystyle \frac{dy}{dx}.\)
Theorem 3.1 Let \(f, g\) be differentiable at \(x_0\). Then \(f \pm g\) and \(fg\) are differentiable at \(x_0\), and \[ (f \pm g)'(x_0) = f'(x_0) \pm g'(x_0), \]
\[ (fg)'(x_0) = g(x_0) f'(x_0) + f(x_0) g'(x_0). \]
If \(g(x_0) \neq 0\), then \[ \left( \frac{f}{g} \right)'(x_0) = \frac{f'(x_0) g(x_0) - f(x_0) g'(x_0)}{[g(x_0)]^2}. \]
Theorem 3.2 Suppose \(f\) is differentiable at \(x_0\), and \(g\) is differentiable at \(y_0 = f(x_0)\). Then the composite function \(g \circ f\) is differentiable at \(x_0\), and \[ \mathrm{d}(g \circ f)(x_0) = \mathrm{d}g(y_0) \circ \mathrm{d}f(x_0). \]
Equivalently,
\[ (g \circ f)'(x_0) = g'(y_0) \cdot f'(x_0) = g'(f(x_0)) f'(x_0). \]
Theorem 3.3 Suppose \(f\) has a continuous inverse function. If \(f\) is differentiable at \(x_0\) and \(f'(x_0) \neq 0\), then \(f^{-1}\) is differentiable at \(y_0 = f(x_0)\), and \[ \mathrm{d}(f^{-1})(y_0) = (\mathrm{d}f(x_0))^{-1}. \]
Equivalently,
\[ (f^{-1})'(y_0) = \frac{1}{f'(x_0)}. \]
Examples
\(d(x^n)=n x^{n-1}dx\qquad(n\in\mathbb{R})\).
\(d(e^x)=e^x\,dx,\qquad d(\ln x)=\frac{1}{x}dx\ (x>0)\).
\(d(a^x)=a^x\ln a\,dx,\qquad d(\log_a x)=\frac{1}{x\ln a}dx\ (x>0)\).
\(d(e^{u})=e^{u}u'dx,\qquad d(\ln u)=\frac{u'}{u}dx\).
\(d(\sin x)=\cos x\,dx,\quad d(\cos x)=-\sin x\,dx,\quad d(\tan x)=\sec^2 x\,dx\).
\(d(\cot x)=-\csc^2 x\,dx,\quad d(\sec x)=\sec x\tan x\,dx,\quad d(\csc x)=-\csc x\cot x\,dx\).
\(d(\arcsin x)=\frac{1}{\sqrt{1-x^2}}dx,\quad d(\arccos x)=-\frac{1}{\sqrt{1-x^2}}dx\).
\(d(\arctan x)=\frac{1}{1+x^2}dx\).
3.2 Higher-Order Derivatives
3.3 Higher-Order Derivative
If a function \(f\) is differentiable everywhere in an interval \(I\), then the function \[ f^{(1)} := f' : I \to \mathbb{R} \] is called the first derivative of \(f\). We denote \(f^{(0)} := f\).
By induction, if the \(n\)-th derivative \(f^{(n)}\) of \(f\) exists, and \(f^{(n)}\) is differentiable at \(x_0\), then we define
\[ f^{(n+1)}(x_0) = \big(f^{(n)}\big)'(x_0). \] We call \(f^{(n+1)}(x_0)\) the \((n+1)\)-th derivative of \(f\) at \(x_0\). The second and third derivatives are denoted by \(f''\) and \(f'''\), respectively.
- We also use the following notations:
\[ f \in \mathcal{C}^n(I) \quad \text{means } f^{(n)} \text{ is continuous on } I; \]
\[ f \in \mathcal{C}^\infty(I) \quad \text{means } f \text{ has derivatives of all orders on } I. \]
Theorem 3.4 Let \(f, g\) have \(n\)-th derivatives at \(x_0\). Then \(\lambda f + \mu g\) and \(fg\) also have \(n\)-th derivatives at \(x_0\), and we have: \[ (\lambda f + \mu g)^{(n)}(x_0) = \lambda f^{(n)}(x_0) + \mu g^{(n)}(x_0) \]
\[ (fg)^{(n)}(x_0) = \sum_{k=0}^{n} C_n^k\, f^{(k)}(x_0)\, g^{(n-k)}(x_0) \]
Theorem 3.5 If \(f\) is \(n\)-times differentiable at \(x_0\), and \(g\) is \(n\)-times differentiable at \(y_0 = f(x_0)\), then the composite function \(g \circ f\) is \(n\)-times differentiable at \(x_0\).
Theorem 3.6 Let \(f, g\) be \(n\)-times differentiable at \(x_0\), and suppose \(g(x_0) \neq 0\). Then \(\dfrac{f}{g}\) is \(n\)-times differentiable at \(x_0\).
Theorem 3.7 The curvature is the reciprocal of the radius of curvature: \[ \kappa = \frac{\Big| \det \begin{pmatrix} x'(t) & x''(t) \\ y'(t) & y''(t) \end{pmatrix} \Big|} {\big[(x'(t))^2 + (y'(t))^2\big]^{3/2}} \]
3.4 Differential Mean Value Theorems
A point \(x_0 \in I\) is called a local maximum of \(f\) if there exists a neighborhood \(V\) of \(x_0\) such that \[ \forall x \in V \cap I,\quad f(x) \le f(x_0). \]
The definition of a local minimum is similar.
A point \(x_0\) is called a critical point of \(f\) if
\[ f'(x_0) = 0. \]
Theorem 3.8 If \(f\) is differentiable at a local extremum \(x_0\), then \(x_0\) is a critical point of \(f\).
Theorem 3.9 (Rolle’s Theorem) Suppose \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a) = f(b)\). Then there exists some \(\xi \in (a,b)\) such that \[ f'(\xi) = 0. \]
Corollary 3.1 Suppose \(f\) is differentiable on the open interval \((a,b)\), and \[ \lim_{x \to a^+} f(x) = \lim_{x \to b^-} f(x) = A \in \mathbb{R} \cup \{\pm\infty\}. \]
Then there exists some \(\xi \in (a,b)\) such that
\[ f'(\xi) = 0. \]
Theorem 3.10 (Cauchy’s Mean Value Theorem) Let \(-\infty \le t_1 < t_2 \le +\infty\), and let \(\alpha, \beta, A, B \in \mathbb{R}\). Suppose \(x(t)\) and \(y(t)\) are differentiable on the open interval \((t_1, t_2)\), and \[ \lim_{t \to t_1^+} x(t) = x_1,\quad \lim_{t \to t_2^-} x(t) = x_2,\quad \lim_{t \to t_1^+} y(t) = y_1,\quad \lim_{t \to t_2^-} y(t) = y_2. \]
Then there exists some \(\xi \in (t_1, t_2)\) such that
\[ x'(\xi)(y_2 - y_1) = y'(\xi)(x_2 - x_1). \]
Theorem 3.11 (Lagrange’s Mean Value Theorem) Suppose \(f\) is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\). Then there exists some \(\xi \in (a,b)\) such that \[ f'(\xi) = \frac{f(b) - f(a)}{b - a}. \]
Theorem 3.12 (Darboux’s Theorem) Suppose \(f\) is differentiable on an interval \(I\). Then \(f'(I)\) is an interval. In particular, if \(x_1, x_2 \in I\) satisfy \[ f'(x_1) < 0 < f'(x_2), \]
then there exists some \(\xi\) between \(x_1\) and \(x_2\) such that
\[ f'(\xi) = 0. \]
Corollary 3.2 If \(f\) is differentiable on an interval \(I\), and \[ f'(x) \ne 0 \quad (\forall x \in I), \]
then \(f\) is strictly monotonic on \(I\).
3.5 Using Derivatives to Analyze Functions
Theorem 3.13 Suppose \(f\) is continuous on an interval \(I\) and differentiable in the interior of \(I\). Then:
\[ \forall x \in I,\quad f'(x) \ge 0\ \ (\text{resp. } f'(x) \le 0) \]
iff \(f\) is non-decreasing (resp. non-increasing) on \(I\).
\[ \forall x \in I,\quad f'(x) = 0 \]
iff \(f\) is constant on \(I\).
\[ \forall x \in I,\quad f'(x) > 0\ \ (\text{resp. } f'(x) < 0) \]
implies that \(f\) is strictly increasing (resp. strictly decreasing) on \(I\).
Theorem 3.14 Assume \(f\) is differentiable at \(x_0\) and \[ f'(x_0) = 0. \]
- If \(f''(x_0) > 0\), then \(x_0\) is a strict local minimum of \(f\).
- If \(f''(x_0) < 0\), then \(x_0\) is a strict local maximum of \(f\).
Theorem 3.15 Assume \(f\) is \(2n\)-times differentiable at \(x_0\), where \(n\) is a positive integer, and \[ f'(x_0) = \cdots = f^{(2n-1)}(x_0) = 0. \]
- If \(f^{(2n)}(x_0) > 0\), then \(x_0\) is a strict local minimum of \(f\).
- If \(f^{(2n)}(x_0) < 0\), then \(x_0\) is a strict local maximum of \(f\).
Theorem 3.16 Assume \(f\) is \((2n+1)\)-times differentiable at \(x_0\), where \(n\) is a positive integer, and \[ f'(x_0) = \cdots = f^{(2n)}(x_0) = 0. \]
- If \(f^{(2n+1)}(x_0) > 0\), then \(f\) is strictly increasing in a neighborhood of \(x_0\).
- If \(f^{(2n+1)}(x_0) < 0\), then \(f\) is strictly decreasing in a neighborhood of \(x_0\).
3.6 Convexity of Functions
A function \(f\) is called convex on an interval \(I\) if for all \(x_1, x_2 \in I\) and all \(t\) with \(0 < t < 1\), we have
\[ f((1-t)x_1 + t x_2) \le (1-t) f(x_1) + t f(x_2). \]
If equality occurs only when \(x_1 = x_2\), then \(f\) is called strictly convex.
Theorem 3.17 Let \(f\) be a convex function on the interval \(I\). Then for any \(x_1, x_2, \cdots, x_n \in I\), and \(\lambda_1, \lambda_2, \cdots, \lambda_n > 0\), with \(\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1\), we have \[ f\left(\sum_{i=1}^{n} \lambda_i x_i\right) \leqslant \sum_{i=1}^{n} \lambda_i f(x_i). \]
If \(f\) is a strictly convex function on \(I\), then when \(x_1, x_2, \cdots, x_n\) are not all equal, we have
\[ f\left(\sum_{i=1}^{n} \lambda_i x_i\right) < \sum_{i=1}^{n} \lambda_i f(x_i). \]
Theorem 3.18 The function \(f\) is a convex function on the interval \(I\) if and only if for any \((x_1, x_2) \subset I\) and any \(x \in (x_1, x_2)\), we have \[ \frac{f(x) - f(x_1)}{x - x_1} \leqslant \frac{f(x_2) - f(x_1)}{x_2 - x_1} \leqslant \frac{f(x_2) - f(x)}{x_2 - x}. \]
Corollary 3.3 Let \(f: I \to \mathbb{R}\) be a convex function. If \(x_0\) is an interior point of \(I\) (i.e., \(x_0 \in \text{int}(I)\)), then \(f\) is continuous at \(x_0\).
Corollary 3.4 Let \(f\) be convex on an open interval \((a, b)\). For every \(x \in (a, b)\), both the left-hand derivative \(f'_-(x)\) and the right-hand derivative \(f'_+(x)\) exist and are finite.
Theorem 3.19 Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b)\), then a necessary and sufficient condition for \(f\) to be a convex function (strictly convex function) on \([a, b]\) is that \(f'\) is increasing (strictly increasing) on \((a, b)\).
3.7 L’Hospital’s Rule
Theorem 3.20 (\(\frac{0}{0}\) Form) Let \(f, g\) be differentiable on the interval \((a,b)\). Assume that \[ f(x) = o(1),\quad g(x)=o(1),\quad x\to a,\quad g'(x)\ne 0. \]
If \[ \lim_{x\to a} \frac{f'(x)}{g'(x)} = A \in \mathbb{R}\cup\{\pm\infty\}, \]
then \[ \lim_{x\to a} \frac{f(x)}{g(x)} = A. \]
Theorem 3.21 (\(\frac{\infty}{\infty}\) Form) Assume that \(g(x)\to\infty\) as \(x\to a\), and \(g'(x)\neq 0\). If \[ \lim_{x\to a} \frac{f'(x)}{g'(x)} = A \in \mathbb{R}\cup\{\pm\infty\}, \]
then \[ \lim_{x\to a} \frac{f(x)}{g(x)} = A. \]
Confirm that \(g'(x)\neq 0\) on a neighborhood of the point
3.8 The Taylor Formula and Its Analytical Applications
Assume that \(f\) has \(n\) derivatives at \(x_0\). The \(n\)-th order Taylor polynomial of \(f\) at \(x_0\) is defined by
\[ T_{f,x_0,n}(h) = f(x_0) + f'(x_0) h + \frac{f''(x_0)}{2} h^2 + \cdots + \frac{f^{(n)}(x_0)}{n!} h^n . \]
We call \(T_{f,x_0,n}(h)\) the \(n\)-th order Taylor polynomial of \(f\) at \(x_0\).
3.8.1 Taylor’s Theorem with Peano Remainder
If \(f\) has \(n\) derivatives at \(x_0\), then a polynomial \[ P_n(h)=a_0+a_1 h+\cdots+a_n h^n \] satisfies \[ f(x_0+h)=P_n(h)+o(h^n),\qquad h\to 0, \] if and only if \(P_n(h)\) is the \(n\)-th order Taylor polynomial of \(f\) at \(x_0\).
3.8.2 Taylor’s Theorem with Lagrange Remainder
Assume that \(f\) is continuous on an interval \(I\) and \((n+1)\) times differentiable on the interior of \(I\). For any interior point \(x_0\in I\) and any \(x\in I\), there exists a point \(\xi\) strictly between \(x\) and \(x_0\) such that
\[ f(x) = T_{f,x_0,n}(x - x_0) + \frac{f^{(n+1)}(\xi)}{(n+1)!}\,(x-x_0)^{\,n+1}. \]
Examples
\[ e^{x} = 1 + x + \frac{x^{2}}{2} + \cdots + \frac{x^{n}}{n!} + o(x^{n}),\qquad x\to 0. \]
\[ \sin x = x - \frac{x^{3}}{3!} + \cdots + (-1)^{n}\frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2}),\qquad x\to 0, \]
\[ \cos x = 1 - \frac{x^{2}}{2!} + \cdots + (-1)^{n}\frac{x^{2n}}{(2n)!} + o(x^{2n+1}),\qquad x\to 0, \]
\[ \ln(1+x) = x- \frac{x^{2}}{2} + \cdots + (-1)^{n-1}\frac{x^{n}}{n} + o(x^{n}),\qquad x\to 0. \]
\[ (1+x)^{\alpha} = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^{2} + \cdots + \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}x^{n} + o(x^{n}),\qquad x\to 0. \]
\[ \arctan x = x - \frac{x^{3}}{3} + \cdots + (-1)^{n}\frac{x^{2n+1}}{2n+1} + o(x^{2n + 2}),\qquad x\to 0, \]