Appendix F — Calculus tutorial

Calculus is the study of the properties of functions. We use calculus to describe the limit behaviour of functions, calculate their rates of change, and calculate the areas under their graphs. In this tutorial, you’ll learn about all these operations through math formulas, interactive visualizations, and code examples.

Click the binder button or this link bit.ly/calctut3 to run this notebooks interactively.

Notebook setup¶

Introduction¶

Example: car trip¶

Suppose you’re driving to see a friend in a neighbouring town. The total length of the journey is 27000 m (27 km) along a straight road. We can describe your position over time as a function $x(t)$, and use the car’s trip odometer to measure the distance travelled. Right before you leave, which we’ll call $t=0$, you reset the trip odometer so it reads $x(0)=0$ m.

Velocity¶

The derivative function $x^{\prime\!}(t)$, pronounced “$x$ prime,” describes how the function $x(t)$ changes over time. In this example $x^{\prime\!}(t)$ is the car’s velocity. If your current velocity is $x^{\prime\!}(t)=15$ m/s, then the car’s position $x(t)$ is increasing by 15 m each second. If you maintain this velocity, the position will increase at a constant rate: $x(0)=0$ m, $x(1)=15$ m, $x(2)=30$ m, and so on until $t=1800$ s when you’ll have travelled $1800\times 15 = 27000$ m and reached your destination.

To estimate the time remaining in the trip, assuming the velocity $x^{\prime\!}(t)$ stays constant, you can divide the remaining distance by the current velocity:

The bigger the derivative $x^{\prime\!}(t)$, the faster you’ll arrive. If you drive two times faster, the time remaining will be halved.

Inverse problem¶

Imagine that the car’s odometer is busted and you don’t have any way to measure the distance $x(t)$ you have travelled over time. The car’s speedometer is still working though, so you know the velocity $x^{\prime\!}(t)$ at all times. Is there a way to calculate the distance travelled using only the information from the speedometer?

There is! You can infer the position at time $t$ from the velocity $x^{\prime\!}(t)$. Think about it---if the speedometer reports $x^{\prime\!}(t)=15$ m/s, then you know that the car’s position is increasing at the rate of 15 m each second. We can describe the total distance travelled until time $t=\tau$ (the Greek letter tau) as the integral of the velocity function $x^{\prime\!}(t)$ between $t=0$ and $t=\tau$:

The integral symbol $\int$ is an elongated $S$ that stands for sum. To calculate the total distance travelled, we imagine splitting the time between $t=0$ and $t=\tau$ into many short time intervals $dt$. During each instant, the position increases by $x^{\prime\!}(t)dt$ m, where $x^{\prime\!}(t)$ is the velocity (measured in m/s), and $dt$ is the time interval (measured in seconds). The integral $\int_{0}^{\tau} x^{\prime\!}(t)\, dt$ calculates the “sum” of these $x^{\prime\!}(t)\,dt$ contributions accumulated between $t=0$ and $t=\tau$.

The situation described in the car example shows that calculus concepts are not theoretical constructs reserved only for math specialists, but common ideas you encounter every day. The derivative $q^{\prime\!}(t)$ describes the rate of change of the quantity $q(t)$. The integral $\int_a^b q(t)dt$ measures the total accumulation of the quantity $q(t)$ during the time period from $t=a$ to $t=b$.

Doing calculus: then and now¶

Symbolic calculations using pen and paper¶

The pen-and-paper approach is a good way to learn calculus because manipulating math symbols “by hand” develops your intuitive understanding of calculus procedures. Writing math on paper allows you to use high-level abstractions and arrive at exact symbolic answers.

Symbolic calculations using SymPy¶

The Python library SymPy allows you to do symbolic math calculations on a computer. We’ll now illustrate the main SymPy capabilities through some examples.

To use SymPy, we start by importing the sympy module under the alias sp, then define the symbolic variable x.

The symbols function creates SymPy symbolic variables. Unlike ordinary Python variables that hold a particular value, SymPy variables act as placeholders that can take on any value. We use symbolic variables to write expressions, just like we do in pen-and-paper calculations.

SymPy is very useful when learning math because it provides functions for common math operations. For example, to factor the expression expr, we can call the function sp.factor.

You can substitute a particular <value> for the variable $\tt{x}$ into the expression expr by calling the method expr.subs() and passing in a python dictionary object {x:<value>}.

We can also also ask SymPy to solve the equation expr = 0, which means to find the values of $x$ that satisfy the equation $4 - x^2 = 0$.

The equation has two solutions so solve returns a list.

Here is another example, where we use sp.solve to solve the quadratic equation $x^2 + 2x - 8 = 0$.

We can check that $x = 2$ and $x = -4$ satisfy the equation $x^2 + 2x - 8 = 0$:

• When $x = 2$, we get $x^2 + 2x - 8 =(2)^2 + 2(2) - 8 = 4 + 4 - 8 = 0$.

• When $x = -4$, we get $x^2 + 2x - 8 =(-4)^2 + 2(-4) - 8 = 16 - 8 - 8 = 0$.

Another useful function is sp.simplify, which you can use to simplify any math expression.

SymPy combined the $5x$ term with the $-3x$ term to obtain the $2x$ term in the simplified output expression.

Sum{y can do many other math operations. We’ll discuss the SymPy functions for doing calculus in the next sections. To learn more about SymPy, I invite you to check out the SymPy tutorial.

Numerical computing using NumPy¶

The Python library NumPy makes it easy to do numerical calculations on a computer. Let’s start by importing the numpy module under the alias np, which is the usual convention.

Her is how e can compute a numerical approximation to $\sqrt{2}$.

NumPy makes it easy to work with arrays of numbers, which are conceptually similar to Python lists, but enable powerful parallel calculations. To give you some idea of this power, we’ll now show create an array of 1001 numbers, and apply a function to each input in this array, to obtain an array of 1001 outputs using just a few lines of code.

We start by calling the NymPy function np.linspace to create the array of smoothly varying inputs. We specify the starting point 0, the end point 2, and number of values we want in the array.

Next, we define a Python function h that corresponds to the math function $h(x) = 4 - x^2$.

We can now apply this function to the array of inputs xs to obtain an array of outputs hxs.

Note that single call to h(xs) applied the function h to all 1001 inputs in the array xs at once. This is called vectorization, and one of the great powers of NumPy: we can write simple math expression that describe what we want to do for each input x, and then NumPy performs these mathematical operations on entire arrays at once, eliminating the need for writing for-loops.

Using NumPy vectorization not only makes numeric code easier to read, but also also makes it run faster! To illustrate this speed up, let’s look at the equivalent Python code for applying h to the list of inputs using Python.

Comparing the timing information output of this cell to the timing information shown earlier, we see that the basic Python version took roughly 10 times longer than the NymPy code.

Scientific computing using SciPy¶

The Python module SciPy is a toolbox of scientific computing helper functions. For example, computing the integral $\int_{0}^2 h(x)dx$, which corresponds to the area under the graph of the function $h(x) = 4 - x^2$ between $x=0$ and $x=2$ requires only two lines of code.

The helper function quad is a powerful tool that the ancients could only dream of.

Applications of calculus¶

We use calculus concepts to describe various quantities in physics, chemistry, biology, engineering, machine learning, business, economics and other domains where quantitative analysis is used. Many laws of nature are expressed in terms of derivatives and integrals, so it’s essential that you learn the language of calculus if you want to study science.

Math prerequisites¶

Let’s start with a review of some key ideas from high school math, which we’ll need as building blocks for calculus.

Sets notation¶

Sets are collections of math objects. Many math ideas are expressed in the language of sets, so it’s worth knowing the notation for sets.

• $\{ \textrm{ definition } \}$: we use curly brackets to define sets. The definition in the curly brackets is either a description of the set’s contents, or a list of the elements in the set.

• $\mathbb{N}$: the set of natural numbers $\mathbb{N} = \{0,1,2,3,4,5,\ldots\}$.

• $\mathbb{N}_+$: the positive natural numbers $\mathbb{N}_+ = \{1,2,3,4,5,\ldots\}$.

• $\mathbb{R}$: the set of real numbers.

• $\mathbb{R}_+$: the set of nonnegative real numbers.

• $x \in S$: this statement is read “$x$ is an element of $S$.” We use this notation to indicate the “type” of the variable $x$. For example, writing “$x \in \mathbb{R}$” tells us $x$ is a real number.

We can use the set-builder notation $\{\, \cdot \; | \; \cdot \, \}$ to define new sets. Inside the curly brackets, we first describe the general kind of mathematical objects we are talking about, followed by the symbol “$|$” (which stands for “such that”), followed by the conditions that identify the elements of the set. For example, the set of nonnegative real numbers $\mathbb{R}_+$ is defined as “all real numbers $x$ such that $x \geq 0$,” which is expressed compactly as $\mathbb{R}_+ = \{ x \in \mathbb{R} \; | \; x \geq 0 \}$ using the set-builder notation.

Python sets¶

The statement $x \in S$ means $x$ is one of the elements of the set $S$.

In Python, we can use the in operator to check if a value is inside the set.

The in statement returns False for a value that are not in the set.

We can perform set operations like union, intersection, and set difference as follows:

The number line¶

The number line is a visual representation of the set of real numbers $\mathbb{R}$

The set of real numbers includes the natural numbers $\{0,1,2,3,\ldots\}$, the integers $\{\ldots, -3,-2,-1,0,1,2,3,\ldots\}$, rational numbers like $-\frac{3}{2}$, 0.5, and $\frac{9}{2}$, as well as irrational numbers like $\sqrt{2}$, $e$, and $\pi$. All the numbers you will run into when doing math can be visualized as a point on the number line.

Infinity¶

The math symbol $\infty$ describes the concept of infinity. We use $\infty$ to represent an infinitely large quantity, that is greater than any number you can think of. Geometrically speaking, you can imagine the number line extends to the right forever towards infinity, as illustrated in the number line figure above. The number line also extends forever to the left, which we denote as negative infinity $-\infty$.

Infinity is not a number but a process. When we use the symbol $+\infty$, we’re describing the process of moving to the right on the number line forever. We go past larger and larger positive numbers and never stop.

Infinity is a key concept in calculus, so it’s important that we develop a precise language to talk about infinitely large numbers and procedures with an infinite number of steps, which we’ll do in the section on limits below.

Functions¶

In Python, we define functions using the def keyword. For example, the code cell below defines the function $f(x) = x^2$, then evaluate it for the input $x=3$.

Function graphs¶

The graph of the function $f(x)$ is obtained by plotting a curve that passes through the set of input-output coordinate pairs $(x, f(x))$, as $x$ varies over a range of inputs.

The function linspace(-3,3,1001) creates an array of 1001 points in the between -3 and 3. We store this sequence of inputs into the variable xs. Next, we compute the output value $f(x)$ for each of the inputs $x$ in the array xs and store the result in the array fxs.

We can now use the function sns.lineplot() to generate the plot.

The helper function plot_func from the ministats library performs the same three steps and produces the same output.

Inverse functions¶

The inverse function $f^{-1}$ performs the inverse operation of the function $f$. If you start from some number $x$, apply $f$, then apply $f^{-1}$, you’ll arrive—full circle—back to the original number $x$:

The inverse of the function $f(x) = x^2$ is the square root function $f^{-1}(x) = \sqrt{x}$. Earlier we computed $f(3) = 9$. If we apply the inverse function to 9, we get $f^{-1}(9) = \sqrt{9} = 3$.

Two other important inverse functions are the exponential function $e^x$ and logarithmic functions $\log(x)$. The logarithmic function $\log(x)$ is the inverse of the exponential function $e^x$, and vice versa. If we compute the exponential of any number, then apply the logarithmic function, we get back the original number.

Similarly, if we take the logarithm of any positive number, then apply the exponential function on the result, we get back to the original number.

Function properties¶

We use the notation $f \colon A \to B$ to denote a function from the input set $A$ to the output set $B$.

• $A$ is the domain of $f$: the set of allowed inputs for the function.

• $B$ is the image of $f$: the set of possible outputs of the function.

For example, the domain of the function $f(x) = x^2$ is $\mathbb{R}$ (any real number) and its image is $\mathbb{R}_+$ (nonnegative real numbers), so we say $f$ is a function of the form $f \colon \mathbb{R} \to \mathbb{R}_+$.

Inventory of functions¶

We’ll now go over a few commonly used math functions and look at their graphs.

Constant function¶

Linear function¶

Quadratic function¶

Polynomial functions¶

Exponential function¶

Absolute value function¶

Square root function¶

Logarithmic function¶

Functions with discrete inputs¶

Later in this tutorial, we’ll study functions with discrete inputs, $a_k : \mathbb{N} \to \mathbb{R}$, which are called sequences. We can express sequences by writing the first few values the sequence $[a_0, a_1, a_2, a_3, \ldots]$, which correspond to evaluating $a_k$ for $k=0$, $k=1$, $k=2$, $k=3$, etc.

Geometry of rectangles and triangles¶

The area of a rectangle of base $b$ and height $h$ is $A = bh$.

The area of a triangle is equal to $\frac{1}{2}$ times the length of its base $b$ times its height $h$: $A = \tfrac{1}{2} b h$.

Trigonometric functions¶

The trigonometric functions $\sin\theta$ and $\cos\theta$ are used in physics and engineering.

Sine function¶

The trigonometric functions np.sin and np.cos take inputs in radians. For example, if you need to compute $\sin(\frac{\pi}{6}\;\textrm{rad})$ you can simplay call the function np.sin on the input np.pi/6:

To compute $\sin(30^\circ)$, you need to fist use the conversion factor of $\frac{\pi\;\textrm{rad}}{180^\circ}$ to convert the angle from degrees to radians. In angle $\theta = 30^\circ$ is $\theta = 30^\circ \frac{\pi\;\textrm{rad}}{180^\circ}$ in radians.

Let’s plot the graph of $f(\theta) = \sin\theta$ over two full cycles.

Cosine function¶

The cosine function is the same as the sine function shifted by $\frac{\pi}{2}$ to the left: $f(\theta) = \cos(\theta) = \sin(\theta+\frac{\pi}{2})$.

Overview and a look ahead¶

The concept map below shows an overview of the calculus ideas you’ll learn in the rest of the tutorial.

Limits¶

We use limits to describe infinitely large quantities, infinitely small quantities, and procedures with infinitely many steps. Limits are an essential theoretical building block for calculus. We use limits to define derivatives, integrals, and series, so this is where we’ll start.

Example 1: Archimedes’ approximation to $\pi$¶

Suppose we want to calculate the area of a circle of radius $r=1$. Today we know the formula for the are of the circle is $A_{\circ} = \pi r^2$, and the area of this circleis $\pi$, but let’s imagine that we’re in ancient times and don’t know the formula.

Sometime around 250 BCE, Archimedes of Syracuse came up with a clever idea to approximate the area of the circle using a regular polygon with $n$ sides inscribed in the circle. The figure below shows the hexagonal (6-sides), octagonal (8-sides), and dodecagonal (12-sides) approximations to the circle.

Part (b) of the figure highlights one of the 16 identical triangular slices for the case $n=8$. The hypotenuse of this triangle has length $r$, the angle $\theta$ is $\frac{360\circ}{16} = \frac{2\pi}{16} = \frac{\pi}{8}\,$ rad, and its sides have length $a=r\cos(\frac{\pi}{8})$ and $b=r\sin(\frac{\pi}{8})$. The area of this triangle is therefore $A_t = \frac{1}{2}ab=\frac{1}{2}r\cos(\frac{\pi}{8})r\sin(\frac{\pi}{8})$, and the area of the octagon is $16A_t = 8r^2\cos(\frac{\pi}{8})\sin(\frac{\pi}{8})$.

More generally, if we draw a polygon with $n$ sides inscribed inside of the circle, the total area of the $2n$ triangles will be:

The $n$-sided-polygon approximation to the area of the circle becomes more and more accurate as $n$ grows. Here is the code for computing the approximations to the area of a circle with an increasing number of sides.

In the limit as $n \to \infty$, the $n$-sided-polygon approximation to the area of the circle will becomes exactly equal to the area

Example 2: Euler’s number¶

A load with annual interest rate of 100% and compounding twice per year will grow by $(1+50\%)(1+50\%) = (1+\frac{1}{2})(1+\frac{1}{2}) = 2.25$ by the end of the year:

If the bank computes the interest three times per year, the amount owed after one year is $(1+\frac{1}{3})(1+\frac{1}{3})(1+\frac{1}{3}) = 2.370$.

With quarterly compounding:

Monthly compounding:

Daily compounding:

Compounding 1000 times per year:

In the limit as $n \to \infty$, the expression $(1+\frac{1}{n})^n$ converges to Euler’s number $e$:

Limits at infinity¶

Limits are useful for describing the behaviour of functions.

Example 3¶

Consider the function $f(x) = \frac{1}{x}$. What happens to $f(x)$ as $x$ goes to infinit? As $x$ becomes larger and larger, the fraction $\frac{1}{x}$ becomes smaller and smaller. In the limit where $x$ goes to infinity, $\frac{1}{x}$ approaches zero: $\lim_{x\to\infty}\frac{1}{x} = 0$. We can see this by looking at the graph of $f(x)$.

Limit formulas¶

The limit of the sum, difference, product, and quotient of two functions are computed as follows:

Example 4¶

Calculate $\lim_{x\to \infty} \frac{2x+1}{x}\,$. We’re given the function $g(x)=\frac{2x+1}{x}$ and must determine what the function looks like for very large values of $x$. We can rewrite the function as $\frac{2x+1}{x}=2+\frac{1}{x}$ then apply the sum formula for limits:

As $x$ goes to infinity, $\frac{1}{x}$ goes to 0, so the second term vanishes, leaving only the 2. We can confirm this visually by looking at the graph of $g(x)$.

Limits at a point¶

We can use limit expressions to describe what happens when a function approaches some number $x=a$. The limit expression $\lim_{x \to a^+} f(x)$ describes the behaviour of the function $f(x)$ as $x$ approaches $a$ from the right. To compute this limit, we imagine a small distance $\delta$ (the Greek letter delta) that that gets smaller and smaller, and compute the following equivalent limit expression

This limit expression describes the value of the function $f$ for inputs like $a+0.1$, $a+0.01$, $a+0.001$, $a+0.0001$, etc. In other words, the input $x$ gets closer and closer to $a$ (from the right) without ever touching $x=a$.

Example¶

Consider the function $f(x) = \frac{1}{x}$ again. The limit $\lim_{x \to 0^+} f(x)$, read “the limit of $f(x)$ as $x$ goes to zero,” asks us to evaluate the function $f$ for inputs like $x=0.1$, $x=0.01$, $x=0.001$, $x=0.0001$, etc. The outputs of $f(x) =~\frac{1}{x}$ get larger and larger as $x$ takes on smaller and smaller values, so the limit goes to infinity:

In words, the function $f(x) = \frac{1}{x}$ “blows up” to infinity as $x$ goes to $0^+$, as shown below.

The limit of $f(x)$ when $x$ approaches $a$ from the left is defined analogously, $\lim_{x\to a^-} f(x) = \lim_{\delta \to 0} f(a - \delta)$. If both limits from the left and from the right at $x=a$ exist and are equal to each other, we can drop the $^+$ and $^-$ annotations, and talk about the limit $x\to a$:

For the two-sided limit of a function to exist at a point, both the limit from the left and the limit from the right must converge to the same number.

Continuity¶

If the function $f(x)$ obeys, $f(a) = \lim_{x\to a} f(x)$, we say the function $f(x)$ is continuous at $x=a$. Geometrically, the graph of the continuous function at $x=a$ is a smooth curve that doesn’t have a hole or a jump at $x=a$.

Example 5¶

Is the function $g(x)=\frac{2x+1}{x}$ continuous at $x=1$? First, we evaluate the function $g(x)$ for the input $x=1$, which gives us $g(1) = \frac{2(1)+1}{1} = \frac{3}{1}=3$. Next we compute the limit from the right at $x=1$ which is:

The $\delta$ terms in the numerator and denominator go to zero. We evaluate the limit from the left using similar steps:

Since the limits from the left and the right are equal, $\lim_{x\to 1^+} g(x) = \lim_{x\to 1^-} g(x)$, we say that the two-sided limit exists $\lim_{x\to 1} g(x) = 3$. Since $\lim_{x\to 1} g(x) = g(3)$, the function $g(x)$ is continuous at $x=1$.

Computing limits using SymPy¶

Arithmetic operations with infinity¶

Infinity $\infty$ is denoted sp.oo (two lowercase os) in SymPy. Infinity is not a number but a process: the process of counting forever. When we use $\infty$ in math expressions and comparisons, we lead to some surprising facts, which we can illustrate using SymPy.

Adding a number to infinity still produces infinity, $\infty + 1 = \infty$.

Infinity is greater than any finite number:

A number divided by infinity is zero, $1/\infty = 0$:

Example 1 using SymPy¶

We can use SymPy to calculate the limit as $n$ goes to infinity:

We see the result of the limit is $\pi r^2$, which is the formula for the area of a circle we were looking for.

Example 2 using SymPy¶

The number $e$ is defined as the limit $\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$:

This limit expression describes the annual growth rate of a loan with a nominal interest rate of 100% and infinitely frequent compounding. Borrow $\$1000$ in such a scheme, and you’ll owe $\$2718.28$ after one year.

Example 3 using SymPy¶

To evaluate the limit from the right or the left at a point, we must provide a fourth argument, either "+" or "-", when calling the function sp.limit.

Example 4 using SymPy¶

Let’s check the result $\lim_{x\to \infty}\!\! \tfrac{2x+1}{x} = 2$.

Example 5 using SymPy¶

Is the function $g(x) = \frac{2x+1}{x}$ continuous at $x=1$?

The fact that all three expressions exist and are equal to each other,

tells us that the function $g(x)=\frac{2x+1}{x}$ is continuous at $x=1$. Intuitively, you can understand continuity by imagining a pen drawing the graph at $x=1$. Th pen comes in from the left at height $\lim_{x\to 1^-} g(x) =~3$, goes through $g(1) = 3$, then leaves to the right at the same height $\lim_{x\to 1^+} g(x) = 3$.

Applications of limits¶

Limits for derivatives¶

The formal definition of a function’s derivative is expressed in terms of the rise-over-run formula for an infinitely short run:

More on derivatives in the next section.

Limit for integrals¶

One way to approximate the area under the graph of the function $f(x)$ between $x=a$ and $x=b$ is to split the area into $n$ rectangular strips of width $\Delta x = \frac{b-a}{n}$ and height $f(x)$, and then calculate the sum of the areas of these rectangles:

We obtain the exact value of the area in the limit where we split the area into an infinite number of rectangles with infinitely narrow width:

Limits for series¶

We use limits to describe the convergence properties of series. For example, the sum of the first $n$ terms of the geometric series $r^k$ corresponds to the following expression:

The series $\sum r^k$ is defined as the limit $n\to \infty$ of the above expression. When $|r|<1$, the geometric series converges to the a finite value:

In each of these domains, limit expressions will help us make precise statements that describe calculus procedures with infinite small lengths and an infinite number of steps.

Derivatives¶

The derivative function, denoted $f'(x)$, $\frac{d}{dx}\big[f(x)\big]$, $\frac{df}{dx}$, or $\frac{dy}{dx}$, describes the instantaneous rate of change of the function $f(x)$. Geometrically, the derivative $f'(x)$ tells us the slope of the function $f$ at $x$.

Let’s start by looking at the average slope calculation between two points separated by the distance $\Delta x$, which is read “delta x.”

The average slope is calculated using the formula $\text{slope} = \frac{\Delta y}{\Delta x}$, where $\Delta x =$ step is the length of the horizontal step, and $\Delta y$ is the change in the function’s output, $\Delta y = f(x+\Delta x) - f(x)$.

Try changing the value of the variable step and rerun the above code. What happens when yo use a very small steps like 0.1, 0.01, and 0.001?

Derivative formula¶

The derivative function $f^{\prime\!}(x)$ is defined as the following limit:

Here $\delta$ plays the same role as $\Delta x$, but we let $\delta$ go to zero.

Numerical derivative calculations¶

The derivative is the slope calculation for a very very short step, which we’ll denote delta to match the math definition.

Derivative formulas¶

Derivative rules¶

In addition to the table of derivative formulas, there are some important derivatives rules that allow you to find derivatives of composite functions.

Constant multiple rule¶

The derivative of $k$ times the function $f(x)$ is equal to $k$ times the derivative of $f(x)$:

Sum rule¶

The derivative of the sum of two functions is the sum of their derivatives:

Product rule¶

The derivative of a product of two functions is the sum of two contributions:

In each term, the derivative of one of the functions is multiplied by the value of the other function.

Quotient rule¶

This formula tells us how to obtain the derivative of a fraction of two functions:

Chain rule¶

If you encounter a situation that includes an inner function and an outer function, like $f(g(x))$, you can obtain the derivative by a two-step process:

In the first step, we leave the inner function $g(x)$ alone and focus on taking the derivative of the outer function $f(x)$. This step gives us $f^{\prime\!}(g(x))$, which is the value of $f^{\prime}$ evaluated at $g(x)$. In the second step, we multiply this expression by the derivative of the inner function $g^{\prime\!}(x)$.

Higher derivatives¶

The second derivative of $f(x)$ is denoted $f^{\prime\prime\!}(x)$ or $\frac{d^2f}{dx^2}$. It is obtained by applying the derivative operation to $f(x)$ twice: $\frac{d}{dx}\big[ \frac{d}{dx}[\texttt{<f>}]\big]$.

Geometrically, the second derivative $f^{\prime\prime\!}(x)$ tells us the curvature of $f(x)$. Positive curvature means the function opens upward and looks like the bottom of a valley. The function $f(x)=x^2$ has derivative $f^{\prime\!}(x) = 2x$ and second derivative $f^{\prime\prime\!}(x) = 2$, which means it has positive curvature everywhere. Negative curvature means the function opens downward and looks like a mountain peak. For example, the function $g(x) = -x^2$ has negative curvature, since its second derivative is $g^{\prime\prime\!}(x) = -2 < 0$.

Examples¶

We’ll use the SymPy function sp.diff to computes the derivative in the following examples.

Example 6¶

Calculate the derivative of $f(x) = e^{x^2}$.

Example 7¶

To find the derivative of $f(x) = \sin(x)e^{x^2}$.

Example 8¶

Calculate the derivative of $\sin(x^2)$.

Checking the derivative formulas using SymPy¶

The table of derivative formulas tells us that $\big[mx+b\big]^\prime = m$. Let’s verify that SymPy gives the same answer.

We can also verify the derivative formula for the polynomial term $\big[x^n\big]^\prime = nx^{n-1}$.

The exponential function $f(x)=e^x$ is an interesting special case, because it is equal to its derivative:

Here is a mix of three other functions from the table of derivative formulas.

Applications of derivatives¶

Tangent lines¶

The tangent line to the function $f(x)$ at $x=x_0$ is the line that passes through the point $(x_0, f(x_0))$ and has the same slope as the function at that point. The tangent line to the function $f(x)$ at the point $x=x_0$ is described by the following formula:

Let’s find the equation of the tangent line to $f(x)=x^2$ at $x_0=1$?

We can see that the tangent line $T_1(x) = 2x -1$ pass through the point $(1,1)$ and has same slope as $f(x)$ at $x=1$.