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

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

Published $\text {1968}$

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$13 \quad$ Derivatives

Definition of a Derivative

If $y = \map f x$, the derivative of $y$ or $\map f x$ with respect to $x$ is defined as:

$13.1$: $\ds \frac {\d y} {\d x} = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = \lim_{\Delta x \mathop \to 0} \frac {\map f {x + \Delta x} - \map f x} {\Delta x}$

where $h = \Delta x$. The derivative is also denoted by $y'$, $d f / d x$ or $\map {f'} x$. 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: $\map {\dfrac \d {\d x} } {u \pm v \pm w \pm \cdots}$

$13.6$: 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: $\dfrac {\d y} {\d x} = \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}$

$13.12$: Derivative of Inverse Function: $\dfrac {\d u} {\d x} = \dfrac 1 {\d x / \d u}$

$13.13$: Corollary of Chain Rule: $\dfrac {\d y} {\d x} = \dfrac {\d y / \d u} {\d x / \d u}$

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 $\sinh u$ with respect to $x$

$13.32$: Derivative of $\cosh u$ with respect to $x$

$13.33$: Derivative of $\tanh u$ with respect to $x$

$13.34$: Derivative of $\coth u$ with respect to $x$

$13.35$: Derivative of $\sech u$ with respect to $x$

$13.36$: Derivative of $\csch u$ with respect to $x$

$13.37$: Derivative of $\sinh^{-1} u$ with respect to $x$

$13.38$: Derivative of $\cosh^{-1} u$ with respect to $x$

$13.39$: Derivative of $\tanh^{-1} u$ with respect to $x$

$13.40$: Derivative of $\coth^{-1} u$ with respect to $x$

$13.41$: Derivative of $\sech^{-1} u$ with respect to $x$

$13.42$: Derivative of $\csch^{-1} u$ with respect to $x$

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 $y = \map f x$ and $\Delta y = \map f {x + \Delta x} - \map f x$. Then:

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

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

$13.50$: $\Delta y = \map {f'} x \Delta x + \epsilon \Delta x$

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 $y$: $\Delta y = \map {f'} x \Delta x + \epsilon \Delta x$

Rules for Differentials

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

$13.52$: $\map \d {u \pm v \pm w \cdots} = \d u \pm \d v \pm \d w \pm \cdots$

$13.53$: $\map \d {u v} = u \rd v + v \rd u$

$13.54$: $\map \d {\dfrac u v} = \dfrac {v \rd u - u \rd v} {v^2}$

$13.55$: $\map \d {u^n} = n u^{n - 1} \rd u$

$13.56$: $\map \d {\sin u} = \cos u \rd u$

$13.57$: $\map \d {\cos u} = -\sin u \rd u$

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:

$13.58-59$: Definition of Partial Derivative

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

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

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

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 Clairaut's Theorem, $13.61.1$ 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:

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

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

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

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