Introduction To Derivatives Lab

Introduction: Start here

In this lab, we will practice our knowledge of derivatives. Remember that our key formula for derivatives, is $f'(x) = \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$. So in driving towards this formula, we will do the following:

  1. Learn how to represent linear and nonlinear functions in code.
  2. Then because our calculation of a derivative relies on seeing the output at an initial value and the output at that value plus delta x, we need an output_at function.
  3. Then we will be able to code the $\Delta f$ function that sees the change in output between the initial x and that initial x plus the $\Delta x$
  4. Finally, we will calculate the derivative at a given x value, derivative_at.

Learning objectives

For this first section, you should be able to answer all of the question with an understanding of our definition of a derivative:

  1. Our intuitive explanation that a derivative is the instantaneous rate of change of a function
  2. Our mathematical definition is that

$f'(x) = \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$

Let's begin: Starting with functions

1. Representing Functions

We are about to learn to take the derivative of a function in code. But before doing so, we need to learn how to express any kind of function in code. This way when we finally write our functions for calculating the derivative, we can use them with both linear and nonlinear functions.

For example, we want to write the function $f(x) = 2x^2 + 4x - 10 $ in a way that allows us to easily determine the exponent of each term.

This is our technique: write the formula as a list of tuples.

A tuple is a list whose elements cannot be reassigned. But everything else, for our purposes, is the same.

tuple = (7, 3)
tuple[0] # 7
tuple[1] # 3

We get a TyperError if we try to reassign the tuple's elements.

tuple[0] = 7
# TypeError: 'tuple' object does not support item assignment

Take the following function as an example:

$$f(x) = 4x^2 + 4x - 10 $$

Here it is as a list of tuples:

four_x_squared_plus_four_x_minus_ten = [(4, 2), (4, 1), (-10, 0)]

So each tuple in the list represents a different term in the function. The first element of the tuple is the term's constant and the second element of the tuple is the term's exponent. Thus $4x^2$ translates to (4, 2) and $-10$ translates to (-10, 0) because $-10$ is the same as $-10*x^0$.

We'll refer to this list of tuples as "list of terms", or list_of_terms.

Ok, so give this a shot. Write $ f(x) = 4x^3 + 11x^2 $ as a list of terms. Assign it to the variable four_x_cubed_plus_eleven_x_squared.

four_x_cubed_plus_eleven_x_squared = [(4, 3), (11, 2)]

2. Evaluating a function at a specific point

Now that we can represent a function in code, let's write a Python function called term_output that can evaluate what a single term equals at a value of $x$.

  • For example, when $x = 2$, the term $3x^2 = 3*2^2 = 12 $.
  • So we represent $3x^2$ in code as (3, 2), and:
  • term_output((3, 2), 2) should return 12
def term_output(term, input_value):
    pass
term_output((3, 2), 2) # 12

Hint: To raise a number to an exponent in python, like 3^2 use the double star, as in:

3**2 # 9 

Now write a function called output_at, when passed a list_of_terms and a value of $x$, calculates the value of the function at that value.

  • For example, we'll use output_at to calculate $f(x) = 3x^2 - 11$.
  • Then output_at([(3, 2), (-11, 0)], 2) should return $f(2) = 3*2^2 - 11 = 1$
def output_at(list_of_terms, x_value):
    pass
three_x_squared_minus_eleven = [(3, 2), (-11, 0)]
output_at(three_x_squared_minus_eleven, 2) # 1 
output_at(three_x_squared_minus_eleven, 3) # 16

Now we can use our output_at function to display our function graphically. We simply declare a list of x_values and then calculate output_at for each of the x_values.

import plotly
from plotly.offline import iplot, init_notebook_mode
init_notebook_mode(connected=True)

from graph import plot, trace_values

x_values = list(range(-30, 30, 1))
y_values = list(map(lambda x: output_at(three_x_squared_minus_eleven, x),x_values))

three_x_squared_minus_eleven_trace  = trace_values(x_values, y_values, mode = 'line')
plot([three_x_squared_minus_eleven_trace], {'title': '3x^2 - 11'})

Moving to derivatives of linear functions

Let's start with a function, $f(x) = 4x + 15$. We represent the function as the following:

four_x_plus_fifteen = [(4, 1), (15, 0)]

We can plot the function by calculating outputs at a range of x values. Note that we use our output_at function to calculate the output at each individual x value.

import plotly
from plotly.offline import iplot, init_notebook_mode
init_notebook_mode(connected=True)

from graph import plot, trace_values, build_layout

x_values = list(range(0, 6))
# layout = build_layout(y_axis = {'range': [0, 35]})


four_x_plus_fifteen_values = list(map(lambda x: output_at(four_x_plus_fifteen, x),x_values))
four_x_plus_fifteen_trace = trace_values(x_values, four_x_plus_fifteen_values, mode = 'line')
plot([four_x_plus_fifteen_trace])

Ok, time for what we are here for, derivatives. Remember that the derivative is the instantaneous rate of change of a function, and is expressed as:

$$ f'(x) = \frac{\Delta f}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x} $$

Writing a function for $\Delta f$

We can see from the formula above that $\Delta f = f(x + \Delta x ) - f(x) $. Write a function called delta_f that, given a list_of_terms, an x_value, and a value $\Delta x $, returns the change in the output over that period.

Hint Don't forget about the output_at function. The output_at function takes a list of terms and an $x$ value and returns the corresponding output. So really output_at is equivalent to $f(x)$, provided a function and a value of x.

four_x_plus_fifteen = [(4, 1), (15, 0)]
def delta_f(list_of_terms, x_value, delta_x):
    pass
delta_f(four_x_plus_fifteen, 2, 1) # 4

So for $f(x) = 4x + 15$, when x = 2, and $\Delta x = 1$, $\Delta f$ is 4.

Plotting our function, delta f, and delta x

Let's show $\Delta f$ and $\Delta x$ graphically.

def delta_f_trace(list_of_terms, x_value, delta_x):
    initial_f_value = output_at(list_of_terms, x_value)
    delta_f_value = delta_f(list_of_terms, x_value, delta_x)
    if initial_f_value and delta_f_value:
        trace =  trace_values(x_values=[x_value + delta_x, x_value + delta_x], 
                              y_values=[initial_f_value, initial_f_value + delta_f_value], mode = 'line',
                              name = 'delta f = ' + str(delta_x))
        return trace
trace_delta_f_four_x_plus_fifteen = delta_f_trace(four_x_plus_fifteen, 2, 1)

Let's add another function that shows the delta x.

def delta_x_trace(list_of_terms, x_value, delta_x):
    initial_f_value = output_at(list_of_terms, x_value)
    if initial_f_value:
        trace = trace_values(x_values=[x_value, x_value + delta_x],
                            y_values=[initial_f_value, initial_f_value], mode = 'line', 
                            name = 'delta x = ' + str(delta_x))
        return trace
from graph import plot, trace_values

trace_delta_x_four_x_plus_fifteen = delta_x_trace(four_x_plus_fifteen, 2, 1)
if four_x_plus_fifteen_trace and trace_delta_f_four_x_plus_fifteen and trace_delta_x_four_x_plus_fifteen:
    plot([four_x_plus_fifteen_trace, trace_delta_f_four_x_plus_fifteen, trace_delta_x_four_x_plus_fifteen], {'title': '4x + 15'})

Calculating the derivative

Write a function, derivative_at that calculates $\frac{\Delta f}{\Delta x}$ when given a list_of_terms, an x_value for the value of $(x)$ the derivative is evaluated at, and delta_x, which represents $\Delta x$.

Let's try this for $f(x) = 4x + 15 $. Round the result to three decimal places.

def derivative_of(list_of_terms, x_value, delta_x):
    pass
derivative_of(four_x_plus_fifteen, 3, 2) # 4.0

We do: Building more plots

Ok, now that we have written a Python function that allows us to plot our list of terms, we can write a function that called derivative_trace that shows the rate of change, or slope, for the function between initial x and initial x plus delta x. We'll walk you through this one.

def derivative_trace(list_of_terms, x_value, line_length = 4, delta_x = .01):
    derivative_at = derivative_of(list_of_terms, x_value, delta_x)
    y = output_at(list_of_terms, x_value)
    if derivative_at and y:
        x_minus = x_value - line_length/2
        x_plus = x_value + line_length/2
        y_minus = y - derivative_at * line_length/2
        y_plus = y + derivative_at * line_length/2
        return trace_values([x_minus, x_value, x_plus],[y_minus, y, y_plus], name = "f' (x) = " + str(derivative_at), mode = 'line')

Our derivative_trace function takes as arguments list_of_terms, x_value, which is where our line should be tangent to our function, line_length as the length of our tangent line, and delta_x which is our $\Delta x$.

The return value of derivative_trace is a dictionary that represents tangent line at that values of $x$. It uses the derivative_of function you wrote above to calculate the slope of the tangent line. Once the slope of the tangent is calculated, we stretch out this tangent line by the line_length provided. The beginning x value is just the midpoint minus the line_length/2 and the ending $x$ value is midpoint plus the line_length/2. Then we calculate our $y$ endpoints by starting at the $y$ along the function, and having them ending at line_length/2*slope in either direction.

tangent_line_four_x_plus_fifteen = derivative_trace(four_x_plus_fifteen, 2, line_length = 4, delta_x = .01)
tangent_line_four_x_plus_fifteen

Now we provide a function that simply returns all three of these traces.

def delta_traces(list_of_terms, x_value, line_length = 4, delta_x = .01):
    tangent = derivative_trace(list_of_terms, x_value, line_length, delta_x)
    delta_f_line = delta_f_trace(list_of_terms, x_value, delta_x)
    delta_x_line = delta_x_trace(list_of_terms, x_value, delta_x)
    return [tangent, delta_f_line, delta_x_line]

Below we can plot our trace of the function as well

delta_x = 1

# derivative_traces(list_of_terms, x_value, line_length = 4, delta_x = .01)

three_x_plus_tangents = delta_traces(four_x_plus_fifteen, 2, line_length= 2*1, delta_x = delta_x)

# only plot the list of traces, if three_x_plus_tangents, does not look like [None, None, None]
if list(filter(None.__ne__, three_x_plus_tangents)):
    plot([four_x_plus_fifteen_trace, *three_x_plus_tangents])

So that function highlights the rate of change is moving at precisely the point x = 2. Sometimes it is useful to see how the derivative is changing across all x values. With linear functions we know that our function is always changing by the same rate, and therefore the rate of change is constant. Let's write functions that allow us to see the function, and the derivative side by side.

from graph import make_subplots, trace_values, plot_figure

def function_values_trace(list_of_terms, x_values):
    function_values = list(map(lambda x: output_at(list_of_terms, x),x_values))
    return trace_values(x_values, function_values, mode = 'line')

def derivative_values_trace(list_of_terms, x_values, delta_x):
    derivative_values = list(map(lambda x: derivative_of(list_of_terms, x, delta_x), x_values))
    return trace_values(x_values, derivative_values, mode = 'line')

def function_and_derivative_trace(list_of_terms, x_values, delta_x):
    traced_function = function_values_trace(list_of_terms, x_values)
    traced_derivative = derivative_values_trace(list_of_terms, x_values, delta_x)
    return make_subplots([traced_function], [traced_derivative])

four_x_plus_fifteen_function_and_derivative = function_and_derivative_trace(four_x_plus_fifteen, list(range(0, 7)), 1)

plot_figure(four_x_plus_fifteen_function_and_derivative)

Summary

In this section, we coded out our function for calculating and plotting the derivative. We started with seeing how we can represent different types of functions. Then we moved onto writing the output_at function which evaluates a provided function at a value of x. We calculated delta_f by subtracting the output at initial x value from the output at that initial x plus delta x. After calculating delta_f, we moved onto our derivative_at function, which simply divided delta_f from delta_x.

In the final section, we introduced some new functions, delta_f_trace and delta_x_trace that plot our deltas on the graph. Then we introduced the derivative_trace function that shows the rate of change, or slope, for the function between initial x and initial x plus delta x.

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