SRSG UNIVERSE-HPC Training

YouTube lecture recording from October 2020

The following YouTube video was recorded for the 2020 iteration of the course. The material is still very similar:

Product, Chain and Quotient Rules

Linear approximation and the derivative

By definition, the derivative of a function $f(x)$ is

$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$

This means that for small $h$ , this expression approximates the derivative. By rearranging this, we have

$f(x + h) \approx f(x) + h f'(x)$

In other words, $f(x+h)$ can be approximated by starting at $f(x)$ and moving a distance $h$ along the tangent $f'(x)$ .

Example

Estimate $\sqrt{5218}$

To do this, we first need an $f(x)$ that is easy to calculate, and close to 5218. For this we can take that $70^2 = 4900$ .

To calculate the approximation, we need $\;f'(x)\;$ , where $\;f(x) = \sqrt{x}\;$ .

$f'(x) = \frac{1}{2 \sqrt{x}}$ > $f'(x) = \frac{1}{2 \sqrt{x}}$

Since 5218 = 4900 + 318, we can set $x = 4900$ and $h = 318$ .

Using the approximation, we have:

$f(5218) \approx f(4900) + 318 \times f'(4900)$ > $f(5218) \approx 70 + 318 \times \frac{1}{140} \approx 72.27$ > $\sqrt{5218} = 72.2357252$ - not a bad appriximation!

Standard derivatives

It's useful to know the derivatives of all the standard functions, and some basic rules.

$\frac{d}{dx} (x^n) = n x^{n-1}$ > $\frac{d}{dx} (\sin x) = \cos x$ > $\frac{d}{dx} (\cos x) = -\sin x$ > $\frac{d}{dx} (e^x) = e^x$

To understand the derivative of sin and cos, consider their graphs, and when they are changing positively (increasing), negatively (decreasing) or not at all (no rate of change).

Graph of sin and cos

Other Differentiation Rules

Differentiation of sums, and scalar multiples

$(f(x) \pm g(x))' = f'(x) \pm g'(x)$ $(a f(x))' = a f'(x)$

Differentiation of products

While differentiating sums, and scalar multiples is straightforward, differentiating products is more complex

$(f(x) g(x) )' \neq f'(x) g'(x)$ > $(f(x) g(x) )' = f'(x) g(x) + g'(x) f(x)$

Example

To illustrate that this works, consider $y = (2x^3 - 1)(3x^3 + 2x)$

If we expand this out, we have that $y = 6x^6 + 4x^4 - 3x^3 - 2x$

From this, clearly, $y' = 36 x^5 + 16x^3 - 9 x^2 - 2$

To use the product rule, instead we say $y = f \times g$ , where $f = 2x^3 - 1$ , and $g = 3x^3 + 2x$ . Therefore

$f'(x) = 6x^2$ > $g'(x) = 9x^2 + 2$ > $y' = f'g + g'f = 6x^2 (3x^3 + 2x) + (9x^2 + 2)(2x^3 - 1)$ > $y' = 18x^5 + 12x^3 + 18x^5 + 4x^3 - 9x^2 - 2 = 36x^5 + 16x^3 - 9x^2 - 2$

So both rules produce the same result. While for simple examples the product rule requires more work, as functions get more complex it saves a lot of time.

Differentiating a function of a function - The Chain Rule

One of the most useful rules is differentiating a function that has another function inside it $y = f(g(x))$ . For this we use the chain rule:

$y = f(g(x))$ > $y'(x) = f'(g(x))\; g'(x) = \frac{df}{dg} \frac{dg}{dx}$

Example 1: $y = (5x^2 + 2)^4$

We can write this as $y = g^4$ , where $g = 5x^2 + 2$ . Given this, we have that

$\frac{dy}{dg} = 4g^3 = 4(5x^2 + 2)^3$ > $\frac{dg}{dx} = 10x$

This means that

$\frac{dy}{dx} = \frac{dy}{dg} \frac{dg}{dx} = 4 (5x^2 + 2)^3 10 x = 40 x (5x^2 + 2)^3$

This extends infinitely to nested functions, meaning $\frac{d}{dx}(a(b(c)) = \frac{d a}{d b} \frac{d}{dx} (b(c)) = \frac{d a}{db} \frac{d b}{dc}\frac{dc}{dx}$

Differentiating the ratio of two functions - The Quotient Rule

If $y(x) = \frac{f(x)}{g(x)}$ , then by using the product rule, and setting $h(x) = (g(x))^{-1}$ , we can show that

$y'(x) = \frac{f'g - g'f}{g^2}$

Example

$y = \frac{3x-1}{4x + 2}$

$f = 3x - 1, \rightarrow f' = 3$ > $g = 4x + 2, \rightarrow g' = 4$ > $y' = \frac{f'g - g'f}{g^2} = \frac{3(4x+2) - 4(3x-1)}{(4x+2)^2}$ > $y' = \frac{12x + 6 - 12 x + 4}{(4x+2)^2} = \frac{10}{(4x+2)^2}$

Differentiating inverses - implicit differentiation

For any function $y = f(x)$ , with a well defined inverse $f^{-1}(x)$ (not to be confused with $(f(x))^{-1})$ ), we have by definition that

$x = f^{-1}(f(x)) = f^{-1}(y)$ .

This means that we can apply the chain rule

$\frac{d}{dx}(x) = \frac{d}{dx}(f^{-1}(y)) = \frac{d}{dy}(f^{-1}(y)) \frac{dy}{dx}$

But since $\frac{d}{dx}(x) = 1$

$\frac{d}{dy}(f^{-1}(y)) = \frac{1}{\frac{dy}{dx}}$

Example: $y = ln(x)$

If $y = ln(x)$ , this means that $f^{-1}(y) = e^y = x$

By definition ( $f^{-1}(y))' = e^y$ , as $e^y$ doesn't change under differentiation. This means that

$\frac{d}{dx}(ln(x)) = \frac{1}{\frac{d}{dy}(f^{-1}(y))} = \frac{1}{e^y}$

But since $y = ln(x)$ :

$\frac{d}{dx}(ln(x)) = \frac{1}{e^{ln(x)}} = \frac{1}{x}$

Example - Differentiating using sympy

In Python, there is a special package for calculating derivatives symbolically, called sympy.

This can quickly and easily calculate derivatives (as well as do all sorts of other analytic calculations).

import sympy as sp

x = sp.symbols('x') # This creates a variable x, which is symbolically represented as the string x.

# Calculate the derivative of x^2
sp.diff(x**2, x)

$\displaystyle 2 x$

sp.diff(sp.cos(x), x)

$\displaystyle - \sin{\left(x \right)}$

f = (x+1)**3 * sp.cos(x**2 - 5)
sp.diff(f,x)

$\displaystyle - 2 x \left(x + 1\right)^{3} \sin{\left(x^{2} - 5 \right)} + 3 \left(x + 1\right)^{2} \cos{\left(x^{2} - 5 \right)}$

f = (x+1)**3 * (x-2)**2 * (x**2 + 4*x + 1)**4
sp.diff(f, x)

$\displaystyle \left(x - 2\right)^{2} \left(x + 1\right)^{3} \cdot \left(8 x + 16\right) \left(x^{2} + 4 x + 1\right)^{3} + 3 \left(x - 2\right)^{2} \left(x + 1\right)^{2} \left(x^{2} + 4 x + 1\right)^{4} + \left(x + 1\right)^{3} \cdot \left(2 x - 4\right) \left(x^{2} + 4 x + 1\right)^{4}$

sp.expand(sp.diff(f, x)) # expand out in polynomial form

$\displaystyle 13 x^{12} + 180 x^{11} + 869 x^{10} + 1250 x^{9} - 2934 x^{8} - 11504 x^{7} - 9142 x^{6} + 10092 x^{5} + 23185 x^{4} + 17068 x^{3} + 6081 x^{2} + 1058 x + 72$

Sympy documentation

You can look at the documentation for Sympy to see many other possibilities (e.g. we will use Sympy to do symbolic integration later on in this course)

https://docs.sympy.org/latest/index.html

Try out Sympy to verify your pen & paper answers to the problem sheets.

Introductory problems

Introductory problems 1

Differentiate the following functions, using the stated rules where indicated:

Product rule: $\displaystyle y=(3x+4x^2)(8+3x^2)$
Product rule: $\displaystyle y=(5x^2-12)(4x^{-1} +2)$
Product rule: $\displaystyle y=x\,\cos x$
Product rule: $\displaystyle y=(3\sqrt{x} + 4x^3)\sin x$
Chain rule: $\displaystyle y=(6x+2)^4$
Chain rule: $\displaystyle y=({5x^3+10})^{1/2}$
Any rules: $\displaystyle y=\sqrt{{1\over{1-3x}}}$
Any rules: $\displaystyle y=(3x+1)\sqrt{5x^2-x}$

Introductory problems 2

Differentiate the following trigonometric functions with respect to $x$ :

$\displaystyle 3\sin(x) + 5\cos(x)$
$\displaystyle \cos(3-2x)$
$\displaystyle \sqrt{\sin(x)}$
$\displaystyle \frac{\cos(x)}{4x}$

Main problems

Main problems 1

Let $y=x^2$ .

Find the exact value of $y$ when $x=2.1$ .
Now estimate the value of $y$ when $x=2.1$ by using the linear approximation formula $f(x_1+h)=f(x_1)+hf'(x_1)$ and letting $x_1=2.0$ and $h=0.1$ .
Compare your estimate to the true value. Which is bigger? What is there about the shape of the graph of $y=x^2$ that accounts for this?
Repeat parts 1. and b) for $x=2.01$ .
Calculate the absolute error in each estimate. How does this error change with the value of $h$ ?

Main problems 2

The energy, $E$ , carried by a photon of wavelength $\lambda$ is given by

$E={hc\over \lambda}$

where $h$ is Planck's constant ( $h=6.63\times 10^{-34}\,\rm{Js}$ ) and $c$ is the speed of light ( $c=3 \times 10^8\,\rm{ms}^{-1}$ ).

Calculate the energy carried by a photon of wavelength 500\thinspace nm.
Sketch a graph to show how E varies with $\lambda$ .
Derive an expression for $\displaystyle \frac{{\rm d}E}{{\rm d}\lambda}$ , and calculate the slope of your graph when $\lambda=500\,\rm{nm}$ .
Hence, or otherwise, estimate the difference in energy between a photon of wavelength $500\,\rm{nm}$ and one of wavelength $505\,\rm{nm}$ .

Main problems 3

On February 10, 1990, the water level in Boston harbour was given by:

$y=5 + 4.9 \cos\left({\pi\over 6} t\right)$

where $t$ is the number of hours since midnight and $y$ is measured in feet.

Sketch y.
Find $\displaystyle \frac{{\rm d}y}{{\rm d}t}$ . What does it represent, in terms of water level?
For $0\le t \le 24$ , when is $\displaystyle \frac{{\rm d}y}{{\rm d}t}$ zero? Explain what it means for $\displaystyle \frac{{\rm d}y}{{\rm d}t}$ to be zero.

Main problems 4

In laminar flow of blood through a cylindrical artery, the resistance $R$ is inversely proportional to the fourth power of the radius $r$ .

Use a linear approximation to show that if $r$ is decreased by 2%, $R$ will increase by approximately 8%.

Main problems 5

Imagine an ocean basin with an inlet on one side. We can model the salinity of the water as a function of the distance from the ocean inlet.

Let $s$ be the water salinity, and $x$ is the distance from the inlet. Then

$\displaystyle s = {s_{0}\alpha X\over(\alpha X - x)}$

where $s_{0}$ is the initial salinity at the inlet, $X$ is the width of the basin, and $\alpha$ is a constant.

Prove that the rate of increase of salinity with distance from the inlet is given by

$\displaystyle \frac{{\rm d}s}{{\rm d}x} = {s\over\alpha X - x}$

Main problems 6

The sine and cosine functions can be written in the form of the following infinite series:

$\sin x = x - {x^3\over3!} + {x^5\over5!} - {x^7\over7!} + \ldots$ > $\cos x = 1 - {x^2\over2!} + {x^4\over4!} - {x^6\over6!} + \ldots$

Differentiate these series term by term to verify the standard expressions for $\displaystyle\frac{{\rm d}}{{\rm d}x}(\sin x)$ and $\displaystyle\frac{{\rm d}}{{\rm d}x}(\cos x)$ .

Extension problems

Extension problems 1

The focal length of the lens of the eye, $f(t)$ , can be controlled so that an object at distance $u(t)$ in front of the eye can be brought to perfect focus on the retina at a constant $v = 1.8\,\rm{cm}$ behind the lens.

A fly is moving towards the eye at a speed of $0.7\,\rm{ms}^{-1}$ .

Assuming that the optics of the eye lens obeys the thin lens formula

${1\over f(t)}={1\over u(t)}+{1\over v},$

find the rate of change of focal length required to keep the fly in perfect focus at a distance of $3\,\rm{m}$ .

Note: consider carefully what you are differentiating with respect to, and the physical interpretation of the mathematics.

Extension problems 2

A contaminated lake is treated with a bactericide.

The rate of change of harmful bacteria $t$ days after treatment is given by

$\frac{{\rm d}N}{{\rm d}t} = -{2000t\over 1+t^2}$

where $N(t)$ is the number of bacteria in $1\,\rm{ml}$ of water.

State with a reason whether the count of bacteria increases or decreases during the period $0\leq t \leq 10$ .
Find the minimum value of $\displaystyle \frac{{\rm d}N}{{\rm d}t}$ during this period.

Extension problems 3

Consider a population of lions $L(t)$ and zebra $Z(t)$ interacting over time $t$ . One type of model for this situation is

$\frac{{\rm d}Z}{{\rm d}t} = aZ - bZL \qquad \rm{and} \qquad \frac{{\rm d}L}{{\rm d}t} = -cL + dZL,$

where $a$ , $b$ , $c$ and $d$ are positive constants.

What values of $\displaystyle \frac{{\rm d}Z}{{\rm d}t}$ and $\displaystyle \frac{{\rm d}L}{{\rm d}t}$ correspond to stable populations?
How would the statement 'zebra go extinct' be represented mathematically?
For parameters $a=0.05$ , $b=0.001$ , $c=0.05$ , and $d=0.00001$ , find all population pairs $(Z,L)$ that yield stable populations. Is extinction inevitable?

Differentiation 2

YouTube lecture recording from October 2020

Product, Chain and Quotient Rules

Linear approximation and the derivative

Example

Standard derivatives

Other Differentiation Rules

Differentiation of sums, and scalar multiples

Differentiation of products

Example

Differentiating a function of a function - The Chain Rule

Example 1: y=(5x2+2)4y = (5x^2 + 2)^4y=(5x2+2)4

Differentiating the ratio of two functions - The Quotient Rule

Example

Differentiating inverses - implicit differentiation

Example: y=ln(x)y = ln(x)y=ln(x)

Example - Differentiating using sympy

Sympy documentation

Introductory problems

Introductory problems 1

Introductory problems 2

Main problems

Main problems 1

Main problems 2

Main problems 3

Main problems 4

Main problems 5

Main problems 6

Extension problems

Extension problems 1

Extension problems 2

Extension problems 3

Example 1: $y = (5x^2 + 2)^4$

Example: $y = ln(x)$