the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

of a function $f (x)$ at $a$ is given by $$\frac{f (a + h) - f (a)}{h} \ \text{or} \ \frac{f (x) - f (a)}{x - a}$$

the process of taking a derivative

instantaneous rate of change

the rate of change of a function at any point along the function $a,$ also called $f' (a),$ or the derivative of the function at $a$

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Calculus Volume 1 · 3.1 Defining the Derivative

Defining the Derivative: practice problems

27 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 1
For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$
$f (x) = 4 x + 7; x_{1} = 2, x_{2} = 5$
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Exercises 3
For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$
$f (x) = x^{2} + 2 x + 1; x_{1} = 3, x_{2} = 3.5$
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Exercises 5
For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$
$f (x) = \frac{4}{3 x - 1}; x_{1} = 1, x_{2} = 3$
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Exercises 7
For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$
$f (x) = x; x_{1} = 1, x_{2} = 16$
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Exercises 9
For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$
$f (x) = x^{1 / 3} + 1; x_{1} = 0, x_{2} = 8$
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Exercises 11
For the following functions,
use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$
$f (x) = 3 - 4 x, a = 2$
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Exercises 13
For the following functions,
use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$
$f (x) = x^{2} + x, a = 1$
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Exercises 15
For the following functions,
use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$
$f (x) = \frac{7}{x}, a = 3$
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Exercises 17
For the following functions,
use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$
$f (x) = 2 - 3 x^{2}, a = - 2$
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Exercises 19
For the following functions,
use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$
$f (x) = \frac{2}{x + 3}, a = - 4$
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Exercises 21
For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.
$f (x) = 5 x + 4, a = - 1$
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Exercises 23
For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.
$f (x) = x^{2} + 9 x, a = 2$
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Exercises 25
For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.
$f (x) = x, a = 4$
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Exercises 27
For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.
$f (x) = \frac{1}{x}, a = 2$
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Exercises 29
For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.
$f (x) = \frac{1}{x ^{3}}, a = 1$
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Exercises 31
For the following exercises, given the function $y = f (x),$
find the slope of the secant line $P Q$ for each point $Q (x, f (x))$ with $x$ value given in the table.
Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
Use the answer from b. to find an equation of the tangent line to $f$ at point $P .$
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Exercises 33
For the following exercises, given the function $y = f (x),$
find the slope of the secant line $P Q$ for each point $Q (x, f (x))$ with $x$ value given in the table.
Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
Use the answer from b. to find an equation of the tangent line to $f$ at point $P .$
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Exercises 35
[T] For the following position functions $y = s (t),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
the simplified expression for the average velocity from $t = 2$ to $t = 2 + h;$
the average velocity between $t = 2$ and where and and
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Exercises 37
[T] For the following position functions $y = s (t),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
the simplified expression for the average velocity from $t = 2$ to $t = 2 + h;$
the average velocity between $t = 2$ and where and and
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Exercises 39
[T] For the following position functions $y = s (t),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
the simplified expression for the average velocity from $t = 2$ to $t = 2 + h;$
the average velocity between $t = 2$ and where and and
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Exercises 41
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
$f (x) = x^{1 / 3}, x = 0$
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Exercises 43
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
$f (x) = {1, x < 1 x, x \geq 1, x = 1$
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Exercises 45
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
[T] The position in feet of a race car along a straight track after $t$ seconds is modeled by the function $s (t) = 8 t^{2} - \frac{1}{16} t^{3} .$
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Exercises 47
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by $s = f (t)$ and $s = g (t),$ where $s$ is measured in feet and $t$ is measured in seconds.
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Exercises 49
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
[T] For the function $f (x) = x^{3} - 2 x^{2} - 11 x + 12,$ do the following.
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Exercises 51
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
Suppose that $N (x)$ computes the number of gallons of gas used by a vehicle traveling $x$ miles. Suppose the vehicle gets $30$ mpg.
Find a mathematical expression for $N (x) .$
What is $N (100)?$ Explain the physical meaning.
What is $N^{'} (100)?$ Explain the physical meaning.
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Exercises 53
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x = a$ for each of the given functions.
[T] For the function $f (x) = \frac{x ^{2}}{x ^{2} + 1},$ do the following.
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Key terms · 4

derivative: the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative
difference quotient: of a function $f (x)$ at $a$ is given by $$\frac{f (a + h) - f (a)}{h} \ \text{or} \ \frac{f (x) - f (a)}{x - a}$$
differentiation: the process of taking a derivative
instantaneous rate of change: the rate of change of a function at any point along the function $a,$ also called $f' (a),$ or the derivative of the function at $a$

Learn 3.1 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 3.1 in the textbook · This section on openstax.org

Defining the Derivative: practice problems

27 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 1

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$

$f (x) = 4 x + 7; x_{1} = 2, x_{2} = 5$

Exercises 3

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$

$f (x) = x^{2} + 2 x + 1; x_{1} = 3, x_{2} = 3.5$

Exercises 5

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$

$f (x) = \frac{4}{3 x - 1}; x_{1} = 1, x_{2} = 3$

Exercises 7

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$

$f (x) = x; x_{1} = 1, x_{2} = 16$

Exercises 9

For the following exercises, use Equation 3.1 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y = f (x) .$

$f (x) = x^{1 / 3} + 1; x_{1} = 0, x_{2} = 8$

Exercises 11

For the following functions,

use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$

$f (x) = 3 - 4 x, a = 2$

Exercises 13

For the following functions,

use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$

$f (x) = x^{2} + x, a = 1$

Exercises 15

For the following functions,

use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$

$f (x) = \frac{7}{x}, a = 3$

Exercises 17

For the following functions,

use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$

$f (x) = 2 - 3 x^{2}, a = - 2$

Exercises 19

For the following functions,

use Equation 3.4 to find the slope of the tangent line $m_{t a n} = f^{'} (a),$ and
find an equation of the tangent line to $f$ at $x = a .$

$f (x) = \frac{2}{x + 3}, a = - 4$

Exercises 21

For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.

$f (x) = 5 x + 4, a = - 1$

Exercises 23

For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.

$f (x) = x^{2} + 9 x, a = 2$

Exercises 25

For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.

$f (x) = x, a = 4$

Exercises 27

For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.

$f (x) = \frac{1}{x}, a = 2$

Exercises 29

For the following functions $y = f (x),$ find $f^{'} (a)$ using Equation 3.5.

$f (x) = \frac{1}{x ^{3}}, a = 1$

Exercises 31

For the following exercises, given the function $y = f (x),$

find the slope of the secant line $P Q$ for each point $Q (x, f (x))$ with $x$ value given in the table.
Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
Use the answer from b. to find an equation of the tangent line to $f$ at point $P .$

Exercises 33

For the following exercises, given the function $y = f (x),$

find the slope of the secant line $P Q$ for each point $Q (x, f (x))$ with $x$ value given in the table.
Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
Use the answer from b. to find an equation of the tangent line to $f$ at point $P .$

Exercises 35

[T] For the following position functions $y = s (t),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find

the simplified expression for the average velocity from $t = 2$ to $t = 2 + h;$
the average velocity between $t = 2$ and where and and

Exercises 37

[T] For the following position functions $y = s (t),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find

the simplified expression for the average velocity from $t = 2$ to $t = 2 + h;$
the average velocity between $t = 2$ and where and and

Exercises 39

[T] For the following position functions $y = s (t),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find

the simplified expression for the average velocity from $t = 2$ to $t = 2 + h;$
the average velocity between $t = 2$ and where and and

Exercises 41