Limits and Derivatives

Abstract

After reading this chapter you know:

• why you need limits and derivatives,

• what derivatives and limits are,

• how to determine a limit,

• how to calculate a derivative,

• what the maximum and minimum of a function are in terms of derivatives and

• what the slope of a function is.

Appendices

Symbols Used in This Chapter (in Order of Their Appearance)

$\underset{x\to a}{\lim \limits }f(x)$	Limit
$\underset{x\to {a}^{+}}{\lim \limits }f(x)$ or $\underset{x\downarrow a}{\lim \limits }f(x)$	Limit from the right
$\underset{x\to {a}^{-}}{\lim \limits }f(x)$or $\underset{x\uparrow a}{\lim \limits }f(x)$	Limit from the left
$\frac{dy}{dx}={y}^{\prime }(x)$	(First order) derivative
$\frac{df(t)}{dt}=\dot{f}$	Time derivative
y ″(x) = y (2)(x)	Second order derivative
y ‴(x) = y (3)(x)	Third order derivative
$\frac{\partial f\left(x,y\right)}{\partial x}$	Partial derivative with respect to x
$\frac{\partial f\left(x,y\right)}{\partial y}$	Partial derivative with respect to y
df(x, y)	Differential

Overview of Equations for Easy Reference

Limit

$$ \underset{x\to a}{\lim \limits }f(x)=L $$

Function with multiple domains

$$ f(x)=\left\{\begin{array}{l}{f}_1(x), for\ x\in \left(-\infty, a\right]\hfill \\ {}{f}_2(x), for\ x\in \left(a,b\right]\hfill \\ {}{f}_3(x), for\ x\in \left(b,\infty \right)\hfill \end{array}\right. $$

Arithmetic rules for limits

• 1. $\underset{x\to c}{\lim \limits }a\dot{\mathrm{c}}f(x)=a\dot{\mathrm{c}}\underset{x\to c}{\lim \limits }f(x)$, when a is constant

• 2. $\underset{x\to c}{\lim \limits}\left[f(x)\pm g(x)\right]=\underset{x\to c}{\lim \limits }f(x)\pm \underset{x\to c}{\lim \limits }g(x)$

• 3. $\underset{x\to c}{\lim \limits}\left[f(x)g(x)\right]=\underset{x\to c}{\lim \limits }f(x)\dot{\mathrm{c}}\underset{x\to c}{\lim \limits }g(x)$

• 4. $\underset{x\to c}{\lim \limits}\left[\frac{f(x)}{g(x)}\right]=\frac{\underset{x\to c}{\lim \limits }f(x)}{\underset{x\to c}{\lim \limits }g(x)}$, if and only if $\underset{x\to c}{\lim \limits }g(x)\ne 0$

• 5. $\underset{x\to c}{\lim \limits }f{(x)}^n={\left[\underset{x\to c}{\lim \limits }f(x)\right]}^n,n\in \mathbb{\mathrm{R}}$

• 6. $\underset{x\to c}{\lim \limits }a=a,$ when a is constant

• 7. $\underset{x\to c}{\lim \limits }x=c$

Heaviside function

$$ f(x)=\left\{\begin{array}{ll}0,\hfill & x\in \left(-\infty, 0\right)\\ {}\frac{1}{2},\hfill & x=0\\ {}1,\hfill & x\in \left(0,\infty \right)\end{array}\right. $$

Special limits

• 1. $\underset{x\to \infty }{\lim \limits }{\left(1+\frac{1}{x}\right)}^x=e$ or $\underset{x\to 0}{\lim \limits }{\left(1+x\right)}^{\frac{1}{x}}=e$

• 2. $\underset{x\to 0}{\lim \limits}\frac{\sin x}{x}=1$

• 3. $\underset{x\to 0}{\lim \limits}\frac{e^x-1}{x}=1$

• 4. $\underset{x\to 0}{\lim \limits}\frac{\ln \left(1+x\right)}{x}=1$

Definition of derivative

$$ \frac{dy}{dx}=\underset{\Delta x\to 0}{\lim \limits}\frac{\Delta y}{\Delta x} $$

Alternative expressions for the derivative if y = f(x)

$$ {y}^{\prime }=\frac{dy}{dx}=\frac{df(x)}{dx}=\frac{d}{dx}f(x)={f}^{\hbox{'}}(x) $$

Basic derivatives

$$ {\displaystyle \begin{array}{ll}y(x)=\mathrm{C},& \frac{dy(x)}{dx}=0,\mathrm{C}\ \mathrm{is}\ \mathrm{constant}\\ {}y(x)={x}^n,\kern0.75em n\in \mathrm{\mathbb{Q}},& \frac{dy(x)}{dx}={nx}^{n-1}\\ {}y(x)={e}^x,& \frac{dy(x)}{dx}={e}^x\\ {}y(x)=\sin x,& \frac{dy(x)}{dx}=\cos x\\ {}y(x)=\cos x,& \frac{dy(x)}{dx}=-\sin x\\ {}y(x)={\mathrm{log}}_ax,& \frac{dy(x)}{dx}=\frac{1}{x\ln a},\left(x>0,a>0\ \mathrm{a}\mathrm{nd}\ \mathrm{a}\ne 1\right)\\ {}\mathrm{y}\left(\mathrm{x}\right)=\ln x,& \frac{dy(x)}{dx}=\frac{1}{x},\kern1em \left(x>0\right)\\ {}y(x)={a}^x,& \frac{dy(x)}{dx}={a}^x\ln a\end{array}} $$

Basic rules for differentiation

$$ {\displaystyle \begin{array}{ll}y(x)= cu(x),c\ \mathrm{is}\ \mathrm{a}\ \mathrm{constant}& {y}^{\prime }(x)={cu}^{\prime }(x)\\ {}y(x)=u(x)+v(x),& {y}^{\prime }(x)={u}^{\prime }(x)+{v}^{\prime }(x)\\ {}y(x)=u(x)v(x),& {y}^{\prime }(x)={u}^{\prime }(x)v(x)+u(x){v}^{\prime }(x),\mathrm{product}\ \mathrm{rule}\\ {}y(x)=\frac{u(x)}{v(x)},& {y}^{\prime }(x)=\frac{u^{\prime }(x)v(x)-u(x){v}^{\prime }(x)\ }{v^2(x)},\mathrm{quotient}\ \mathrm{rule}\\ {}y(x)=y(u)u=u(x),& \frac{dy}{du}\frac{du}{dx},\mathrm{chain}\ \mathrm{rule}\\ {}y=y(v)\kern0.5em v=v(u)\kern0.5em u=u(x),& \frac{dy}{dx}=\frac{dy}{dv}\frac{dv}{du}\frac{du}{dx},\mathrm{chain}\ \mathrm{rule}\end{array}} $$

Higher order derivatives.

Second order derivative

$$ {y}^{{\prime\prime} }(x)={y}^{(2)}(x)=\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right) $$

Third order derivative

$$ {y}^{{\prime\prime\prime} }(x)={y}^{(3)}(x)=\frac{d^3y}{dx^3}=\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)=\frac{d}{dx}\left[\frac{d}{dx}\left(\frac{dy}{dx}\right)\right] $$

Partial derivatives.

First order partial derivatives for a function of two variables:

$$ \frac{\partial f\left(x,y\right)}{\partial x}={\left(\frac{df\left(x,y\right)}{dx}\right)}_{y= constant} $$

$$ \frac{\partial f\left(x,y\right)}{\partial y}={\left(\frac{df\left(x,y\right)}{dy}\right)}_{x= constant} $$

Second order partial derivatives

$$ \frac{\partial^2f\left(x,y,\dots \right)}{\partial {x}^2}=\frac{\partial }{\partial x}\left(\frac{\partial f\left(x,y,..\right)}{\partial x}\right) $$

$$ \frac{\partial^2f\left(x,y,\dots \right)}{\partial {y}^2}=\frac{\partial }{\partial y}\left(\frac{\partial f\left(x,y,..\right)}{\partial y}\right) $$

$$ \frac{\partial^2f\left(x,y,\dots \right)}{\partial x\partial y}=\frac{\partial }{\partial x}\left(\frac{\partial f\left(x,y,..\right)}{\partial y}\right) $$

More notations for partial derivatives

$$ \frac{\partial f\left(x,y,..\right)}{\partial x}=\frac{\partial }{\partial x}f\left(x,y,..\right)={f}_x\left(x,y,\dots \right)={\partial}_xf\left(x,y,\dots \right) $$

Total differential of a function of two variables

$$ df\left(x,y\right)=\frac{\partial f\left(x,y\right)}{\partial x} dx+\frac{\partial f\left(x,y\right)}{\partial y} dy $$

Answers to Exercises

• 6.1.

  • a) 18

  • b) 12

  • c) 1.5

  • d) Does not exist.

• 6.2.

  • a) −1

  • b) −5

  • c) 5/6

  • d) −1/2

• 6.3.

  • a) ∞

  • b) 0

  • c) 0

  • d) ½

  • e) ∞

• 6.4.

  • a) y′ = 6x + 6x ²

  • b) y′ = 3  cos 3x

  • c) y′ = 22x ²¹

  • d) y′ = 7

  • e) y′ =  − 363  sin 11x

  • f) y′ =  − e ^−x

  • g) ${y}^{\prime }=12{x}^3-\frac{1}{2\sqrt{x}}$

• 6.5.

  • a) y′ = cos x

  • b) $y{}^{\prime }={e}^x\left({x}^{\frac{1}{4}}+\frac{1}{4}{x}^{-\frac{3}{4}}\right)$

  • c) y′ = a ^x x ^a − 1(x ln  a + a)

  • d) $y{}^{\prime }=\frac{5{x}^2-1}{2\sqrt{x}}$

  • e) $y{}^{\prime }=\frac{6}{5}{x}^{\frac{3}{5}}-\frac{3}{2}\sqrt{x}$

  • f) y′ = 10^x(1 + x  ln 10)

  • g) y′ = e ^ax sin x(1 + a ²)

• 6.6.

  • a) y′(t) =  − iωe ^−iωt, y″(t) =  − ω ² e ^−iωt, y‴(t) = iω ³ e ^−iωt

  • b) y′(x) = 3  cos  3x, y″(x) =  − 9  sin  3x, y‴(x) =  − 27 cos 3x

  • c) y′(x) = 22x ²¹, y″(x) = 462x ²⁰, y‴(x) = 9240x ¹⁹

  • d) y′(x) = x ⁻²(− ln  x + 1), y″(x) = x ⁻³(2  ln  x − 3), y‴(x) = x ⁻⁴(−6  ln  x + 11)

• 6.7. $\frac{\partial f\left(x,t\right)}{\partial x}=-{ite}^{- ixt}$, $\frac{\partial f\left(x,t\right)}{\partial t}=-{ixe}^{- ixt}$, $\frac{\partial^2f\left(x,t\right)}{\partial {x}^2}=-{t}^2{e}^{- ixt}$, $\frac{\partial^2f\left(x,t\right)}{\partial {t}^2}=-{x}^2{e}^{- ixt}$, $\frac{\partial^2f\left(x,t\right)}{\partial x\partial t}=-{e}^{- ixt}\left(i+ xt\right)$ _.

Glossary

Absolute

As in absolute maximum or minimum: the largest maximum or minimum value over the entire domain of a function.

Argument

Input variable of a function.

Asymptote

A line or a curve that approaches a given curve arbitrarily closely; their point of touch tends towards infinity. It is the tangent of a curve at infinity.

Asymptotic discontinuity

A function that has a vertical asymptote for the argument belonging to the discontinuity.

Blood oxygenation level

The level of oxygen in the blood.

Continuous

A function that is defined on its domain, for which sufficiently small changes in the input result in arbitrarily small changes in the output.

Derivative

Rate of change of a function; also the slope of the tangent line.

Differential

Infinitesimal differences of a function.

Discrete

Opposite of continuous; a discrete variable can only take on specific values.

Domain

The set of arguments for which a function is defined.

Extrema

Collective name for maxima and minima of a function.

Hemodynamic response

Here: the increase in blood flow to active brain neuronal tissue.

Infinity

An abstract concept that in the context of mathematics can be thought of as a number larger than any number.

Limit

A limit is the value that a function or sequence “approaches” as the variable approaches some value.

Local extrema

As in local maximum or minimum: a maximum or minimum value of the function in a neighbourhood of the point.

One-sided limit

Limit that only exists for the variable of a function approaching some value from one side.

Optimisation problem

The problem of finding the best solution from all possible solutions.

Partial derivative

A derivative of a function of several variables with respect to one variable, considering remaining variables as a constant.

Piecewise

A function that is defined differently for different parts of its domain.

Point of inflection

Point where the curvature changes sign, i.e. where the derivative has an extrema.

Propagation

Movement.

Regressor

An independent variable that can explain a dependent variable in a regression model.

Removable discontinuity

(Also known as point discontinuity) discontinuity that occurs when a function is defined differently at a single point, or when a function is not defined at a certain point.

Root

Point where a function is equal to zero.

(To) sample

To digitize an analog signal, analog-to-digital (A/D) conversion.

Stationary point

A point on a curve where the derivative is equal to zero.

Tangent (line)

A straight line that touches a function at only one point, also an instantaneous rate of change at that point.