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 $y = \map f x$, the derivative of $y$ or $\map f x$ with respect to $x$ is defined as:
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.
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 $y = \map f x$ and $\Delta y = \map f {x + \Delta x} - \map f x$. Then:
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:
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:
- $13.58$: $\dfrac {\partial f} {\partial x} = \ds \lim_{\Delta x \mathop \to 0} \dfrac {\map f {x + \Delta x, y} - \map f {x, y} } {\Delta x}$
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:
where $\d x = \Delta x$ and $\d y = \Delta y$.
Extension to functions of more than two variables are exactly analogous.