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

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

Published $1968$.


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$13 \quad$ 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:

$13.1$: $\displaystyle \frac {\mathrm d y} {\mathrm d x} = \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h = \lim_{\Delta x \to 0} \frac {f \left({x + \Delta x}\right) - f \left({x}\right)} {\Delta x}$

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.

$13.2$: Derivative of $c$
$13.3$: Derivative of $c x$
$13.4$: Derivative of $c x^n$
$13.5$: Derivative of Sum of Functions: $\dfrac {\mathrm d} {\mathrm d x} \left({u \pm v \pm w \pm \cdots}\right)$
$13.6$: Derivative of $c u$
$13.7$: Derivative of $u v$
$13.8$: Derivative of $u v w$
$13.9$: Derivative of $\dfrac u v$
$13.10$: Derivative of $u^n$
$13.11$: Chain rule: $\dfrac {\mathrm d y} {\mathrm d x} = \dfrac {\mathrm d y} {\mathrm d u} \dfrac {\mathrm d u} {\mathrm d x}$
$13.12$: Derivative of Inverse Function: $\dfrac {\mathrm d u} {\mathrm d x} = \dfrac 1 {\mathrm d x / \mathrm d u}$
$13.13$: Corollary of Chain Rule: $\dfrac {\mathrm d y} {\mathrm d x} = \dfrac {\mathrm d y / \mathrm d u} {\mathrm d x / \mathrm d u}$


Derivatives of Trigonometric and Inverse Trigonometric Functions

$13.14$: Derivative of $\sin x$
$13.15$: Derivative of $\cos x$
$13.16$: Derivative of $\tan x$
$13.17$: Derivative of $\cot x$
$13.18$: Derivative of $\sec x$
$13.19$: Derivative of $\csc x$
$13.20$: Derivative of $\sin^{-1} x$
$13.21$: Derivative of $\cos^{-1} x$
$13.22$: Derivative of $\tan^{-1} x$
$13.23$: Derivative of $\cot^{-1} x$
$13.24$: Derivative of $\sec^{-1} x$
$13.25$: Derivative of $\csc^{-1} x$


Derivatives of Exponential and Logarithmic Functions

$13.26$: Derivative of $\log_a x$
$13.27$: Derivative of $\ln x$
$13.28$: Derivative of $a^x$
$13.29$: Derivative of $e^x$
$13.30$: Derivative of $u^v$


Derivatives of Hyperbolic and Inverse Hyperbolic Functions

$13.31$: Derivative of $\sinh x$
$13.32$: Derivative of $\cosh x$
$13.33$: Derivative of $\tanh x$
$13.34$: Derivative of $\coth x$
$13.35$: Derivative of $\operatorname{sech} x$
$13.36$: Derivative of $\operatorname{csch} x$
$13.37$: Derivative of $\sinh^{-1} x$
$13.38$: Derivative of $\cosh^{-1} x$
$13.39$: Derivative of $\tanh^{-1} x$
$13.40$: Derivative of $\coth^{-1} x$
$13.41$: Derivative of $\operatorname{sech}^{-1} x$
$13.42$: Derivative of $\operatorname{csch}^{-1} x$


Higher Derivatives

$13.43$: Definition of Second Derivative
$13.44$: Definition of Third Derivative
$13.45$: Definition of $n$th Derivative


Leibnitz's Rule for Higher Derivatives of Products

$13.46$: $n$th Derivative of Product
$13.47$: Second Derivative of Product
$13.48$: Third Derivative of Product


Differentials

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

$13.49$: Definition of Differential: $\dfrac {\Delta y} {\Delta x} = \dfrac {f \left({x + \Delta x}\right) - f \left({x}\right)} {\Delta x} = f' \left({x}\right) + \epsilon = \dfrac {\d y} {\d x} + \epsilon$

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:

$13.51$: Definition of Differential of $y$: $\Delta y = f' \left({x}\right) \Delta x + \epsilon \Delta x$


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

$13.58-59$: Definition of Partial Derivative
$13.58$: $\dfrac {\partial f} {\partial x} = \lim_{\Delta x \mathop \to 0} \dfrac {f \left({x + \Delta x, y}\right) - f \left({x, y}\right)} {\Delta x}$

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

$13.59$: $\dfrac {\partial f} {\partial y} = \lim_{\Delta y \mathop \to 0} \dfrac {f \left({x, y + \Delta y}\right) - f \left({x, y}\right)} {\Delta y}$


Partial derivatives of higher order can be defined as follows.

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

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 $f \left({x,y}\right)$ is defined as:

$13.62$: Definition of Differential for Multi-Variable Functions: $\mathrm d f = \dfrac {\partial f} {\partial x} \mathrm d x + \dfrac {\partial f} {\partial y} \mathrm d y$

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

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


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