We can find the derivative of a graph of a function by drawing tangent lines on the function’s graph. In the activity below, you try this process out for yourself.

To get an idea of what we are going to try to do, read through Question 1 in Section 11.4.

Our goal in this activity is to find the slopes of tangent lines at several x values. The handout below has four pages. On the first and third pages contain the graphs along with a table of x values. Pages two and four contain empty graphs.

Print out this PDF file. To fill out the tables on pages one and three, draw tangent lines on the x values indicated in the tables. Now use the grid to estimate the slope of these tangent lines. Enter those values in the second column of the table.

Once you have filled out the table, graph the ordered pairs on the corresponding blank graph. In Example 1, you should see a very obvious pattern ( a line) to the data points you just graphed. In Example 2, there is also a pattern (a parabola) although it may not be as obvious.

Each of the ordered pairs in the table gives the derivative of the graph you took it off of. The video below illustrates this process.

This technique of drawing the derivative is not a very effective method for finding the derivative of a function. It gives us the graph, but not necessarily the formula. In the rest of Section 11.4, you’ll learn how to find the derivative using the definition with limits. You will also learn some shortcuts to take the derivative.

A tangent line to a function is a line that looks most like the function at a point. In common terms, it just grazes the function.

To find its equation, we need to locate the point where the two meet as well as the slope of the function at that point. Then we can use the slope-intercept form or point-slope form of a line to get the equation.

Problem Find the equation of the tangent line to

atx = 3

Solution Since this problem is asking for the equation of a line, let’s start with the point-slope form

This requires a point (x_{1}, y_{1}) and slope m. We’ll use the function to get the point and the derivative to get the slope of the tangent line.

Find the point: We are given a pointx = 3. To find the corresponding y value, put the x value into the function

Find the slope of the tangent line: We need h′(3) to get the slope of the tangent line. We’ll use the Power Rule to take the derivative,

The slope of the tangent is

Write the equation of the tangent line: Putting the point (3, 10) and the slope 9 into the line yields

If you are asked to write this in slope-intercept form, you’ll need to solve this for y to give

If you graph h(x) and the tangent line together, it should be obvious that your tangent line is correct (ie. tangent).

The most difficult part of finding a derivative is evaluating the limit involved in the definition of the derivative at a point. Often there is some algebra and simplifying involved as the example below demonstrates.

Problem Suppose the function g(x) is given by

Use the definition of the derivative at a point to compute g´(3).

Solution The definition of the derivative of g(x) at x = 3 is

The function value g(3) is calculated to be

The function value g(3 + h) must be calculated carefully.

Form the difference quotient and simplify:

The derivative is completed by taking the limit ash approaches zero,

The derivative of g(x) evaluated at x = 3,g´(3), is 11.

A tangent line to a function is a line that looks most like the function at a point. In common terms, it just grazes the function.

To find its equation, we need to locate the point where the two meet as well as the slope of the function at that point. Then we can use the slope-intercept form or point-slope form of a line to get the equation.

Find the equation of the tangent line to

at x = 3.

Since this problem is asking for the equation of a line, let’s start with the point-slope form

This requires a point (x_{1}, y_{1}) and slope m. We’ll use the function to get the point and the derivative to get the slope of the tangent line.

Find the point: We are given a point x = 3. To find the corresponding y value, put the x value into the function

Find the slope of the tangent line: We need h′(3) to get the slope of the tangent line. We’ll use the Power Rule to take the derivative,

The slope of the tangent is

Write the equation of the tangent line: Putting the point (3, 10) and the slope 9 into the line yields

If you are asked to write this in slope-intercept form, you’ll need to solve this for y to give

If you graph h(x) and the tangent line together, it should be obvious that your tangent line is correct (ie. tangent).

In many business problems, we are interested in knowing the rate at which some quantity is changing. If this quantity is modeled by a function, the rate is modeled by the derivative of the function.

In the problem below, we are given a model of revenue for Verizon, a national wireless carrier. To find the rate, we’ll need to take the derivative of the revenue function.

From 2006 through 2011, Verizon Wireless grew steadily. As the number of connections increased, so did the revenue from those connections.A cubic model for the revenue over this period is

where c is the number of connections in millions.

At what rate is the revenue increasing when there are 80,000,000 connections?

Questions that ask about rate or refer to “marginal” are questions about the derivative. In this case, it is referring to the value of the derivative at c = 80. The derivative of the revenue function is

The value of the derivative at 80 million connections is

To find the units on this rate, recall that the derivative is also the slope of the tangent line at c = 80. The slope of this line has vertical units of billions of dollars and horizontal units of millions of connections. This may be simplified to

Since a billion divided by a million is one thousand. So means that an additional connection at this level results in an increase in revenue of 0.31265 thouusand dollars or $321.65.