Derivatives and Differentiation

Definition

A point $x_0$ is called an interior point of $I$, if there exists $\delta > 0$ such that

$$\begin{equation*} \forall x,\ |x - x_0| < \delta \Rightarrow x \in I. \end{equation*}$$

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

$$\begin{equation*} f(x_0 + h) = f(x_0) + L(h) + o(h), \quad h \to 0. \end{equation*}$$

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 $x_0$.

$$\begin{equation*} df(x_0)(h) = f'(x_0)h. \end{equation*}$$ $$\begin{equation*} \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0) - f'(x_0)h}{h} = \lim_{h \to 0} \frac{o(h)}{h} = 0. \end{equation*}$$

The derivative is defined as:

$$\begin{equation*} f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}. \end{equation*}$$

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

$$\begin{equation*} f'(x_0) = \lim_{h \to 0} \frac{\Delta y}{\Delta x} = \left. \frac{dy}{dx} \right|_{x_0}. \end{equation*}$$

• Lagrange notation: $f'(x_0)$
• Leibniz notation: $\displaystyle \frac{dy}{dx}$

Chain Rule

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

$$\begin{equation*} \mathrm{d}(g \circ f)(x_0) = \mathrm{d}g(y_0) \circ \mathrm{d}f(x_0). \end{equation*}$$

Equivalently,

$$\begin{equation*} (g \circ f)'(x_0) = g'(y_0) \cdot f'(x_0) = g'(f(x_0)) f'(x_0). \end{equation*}$$

Differentiability of the Inverse Function

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

$$\begin{equation*} \mathrm{d}(f^{-1})(y_0) = (\mathrm{d}f(x_0))^{-1}. \end{equation*}$$

Equivalently,

$$\begin{equation*} (f^{-1})'(y_0) = \frac{1}{f'(x_0)}. \end{equation*}$$

Four Arithmetic Operations of Differentiation

Let $f, g$ be differentiable at $x_0$. Then $f \pm g$ and $fg$ are differentiable at $x_0$, and

$$\begin{equation*} (f \pm g)'(x_0) = f'(x_0) \pm g'(x_0), \end{equation*}$$ $$\begin{equation*} (fg)'(x_0) = g(x_0) f'(x_0) + f(x_0) g'(x_0). \end{equation*}$$

If $g(x_0) \neq 0$, then

$$\begin{equation*} \left( \frac{f}{g} \right)'(x_0) = \frac{f'(x_0) g(x_0) - f(x_0) g'(x_0)}{[g(x_0)]^2}. \end{equation*}$$

Examples:

1. $d(x^n)=n x^{n-1}dx\qquad(n\in\mathbb{R})$.

2. $d(e^x)=e^x\,dx,\qquad d(\ln x)=\frac{1}{x}dx\ (x>0)$.

3. $d(a^x)=a^x\ln(a)\,dx,\qquad d(\log_a x)=\frac{1}{x\ln a}dx\ (x>0)$.

4. $d(e^{u})=e^{u}u'dx,\qquad d(\ln u)=\frac{u'}{u}dx$.

5. $d(\sin x)=\cos x\,dx,\quad d(\cos x)=-\sin x\,dx,\quad d(\tan x)=\sec^2 x\,dx$.

6. $d(\cot x)=-\csc^2 x\,dx,\quad d(\sec x)=\sec x\tan x\,dx,\quad d(\csc x)=-\csc x\cot x\,dx$.

7. $d(\arcsin x)=\frac{1}{\sqrt{1-x^2}}dx,\quad d(\arccos x)=-\frac{1}{\sqrt{1-x^2}}dx$.

8. $d(\arctan x)=\frac{1}{1+x^2}dx$.