# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 13

## Murray R. Spiegel: *Mathematical Handbook of Formulas and Tables: Chapter 13*

Published $\text {1968}$.

## $13 \quad$ Derivatives

### Definition of a Derivative

If $y = f \left({x}\right)$, the derivative of $y$ or $f \left({x}\right)$ with respect to $x$ is defined as:

where $h = \Delta x$. The derivative is also denoted by $y'$, $d f / d x$ or $f ' \left({x}\right)$. The process of taking a derivative is called differentiation.

### General Rules of Differentiation

In the following, $u, v, w$ are functions of $x$; $a, b, c, n$ any constants, restricted if indicated; $e = 2.71828 \ldots$ is the natural base of logarithms; $\ln u$ denotes the natural logarithm of $u$ where it is assumed that $u > 0$ and all angles are in radians.

### Derivatives of Trigonometric and Inverse Trigonometric Functions

### Derivatives of Exponential and Logarithmic Functions

### Derivatives of Hyperbolic and Inverse Hyperbolic Functions

### Higher Derivatives

### Leibnitz's Rule for Higher Derivatives of Products

### Differentials

Let $y = f \left({x}\right)$ and $\Delta y = f \left({x + \Delta x}\right) - f \left({x}\right).$ Then:

where $\epsilon \to 0$ as $\Delta x \to 0$. Thus:

- $13.50$: $\Delta y = f' \left({x}\right) \Delta x + \epsilon \Delta x$

If we call $\Delta x = \d x$ the differential of $x$, then we define the differential of $y$ to be:

### Rules for Differentials

The rules for differentials are exactly analogous to those for derivatives.

- $13.52$: $\mathrm d \left({u \pm v \pm w \cdots}\right) = \mathrm d u \pm \mathrm d v \pm \mathrm d w \pm \cdots$

- $13.53$: $\mathrm d \left({u v}\right) = u \ \mathrm d v + v \ \mathrm d u$

- $13.54$: $\mathrm d \left({\dfrac u v}\right) = \dfrac{v \ \mathrm d u - u \ \mathrm d v} {v^2}$

- $13.55$: $\mathrm d \left({u^n}\right) = n u^{n - 1} \ \mathrm d u$

- $13.56$: $\mathrm d \left({\sin u}\right) = \cos u \ \mathrm d u$

- $13.57$: $\mathrm d \left({\cos u}\right) = - \sin u \ \mathrm d u$

### Partial Derivatives

Let $\map f {x, y}$ be a function of the two variables $x$ and $y$. Then we define the partial derivative of $\map f {x, y}$ with respect to $x$, keeping $y$ constant, to be:

- $13.58$: $\dfrac {\partial f} {\partial x} = \displaystyle \lim_{\Delta x \mathop \to 0} \dfrac {\map f {x + \Delta x, y} - \map f {x, y} } {\Delta x}$

Similarly the partial derivative of $\map f {x, y}$ with respect to $y$, keeping $x$ constant, is defined to be:

- $13.59$: $\dfrac {\partial f} {\partial y} = \displaystyle \lim_{\Delta y \mathop \to 0} \dfrac {\map f {x, y + \Delta y} - \map f {x, y} } {\Delta y}$

Partial derivatives of higher order can be defined as follows.

- $13.60-61$: Definition of Second Partial Derivative
- $13.60.1$: $\dfrac {\partial^2 f} {\partial x^2} = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }$
- $13.60.2$: $\dfrac {\partial^2 f} {\partial y^2} = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }$
- $13.61.1$: $\dfrac {\partial^2 f} {\partial x \partial y} = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }$
- $13.61.2$: $\dfrac {\partial^2 f} {\partial y \partial x} = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }$

By Partial Differentiation Operator is Commutative for Continuous Functions, $13.61.1$ and $13.61.2$ will be equal if the function and its partial derivatives are continuous, that is, in such case the order of differentiation makes no difference.

The differential of $\map f {x, y}$ is defined as:

where $\d x = \Delta x$ and $\d y = \Delta y$.

Extension to functions of more than two variables are exactly analogous.