gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

a function for which $f' (a)$ exists is differentiable at $a$

a function for which $f' (x)$ exists for each $x$ in the open set $S$ is differentiable on $S$

differentiable function

a function for which $f' (x)$ exists is a differentiable function

higher-order derivative

a derivative of a derivative, from the second derivative to the nth derivative, is called a higher-order derivative

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Calculus Volume 1 · 3.2 The Derivative as a Function

The Derivative as a Function: practice problems

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

Exercises 55
For the following exercises, use the definition of a derivative to find $f^{'} (x) .$
$f (x) = 2 - 3 x$
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Exercises 57
For the following exercises, use the definition of a derivative to find $f^{'} (x) .$
$f (x) = 4 x^{2}$
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Exercises 59
For the following exercises, use the definition of a derivative to find $f^{'} (x) .$
$f (x) = 2 x$
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Exercises 61
For the following exercises, use the definition of a derivative to find $f^{'} (x) .$
$f (x) = \frac{9}{x}$
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Exercises 63
For the following exercises, use the definition of a derivative to find $f^{'} (x) .$
$f (x) = \frac{1}{x}$
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Exercises 65
For the following exercises, use the graph of $y = f (x)$ to sketch the graph of its derivative $f^{'} (x) .$
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Exercises 67
For the following exercises, use the graph of $y = f (x)$ to sketch the graph of its derivative $f^{'} (x) .$
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Exercises 69
For the following exercises, the given limit represents the derivative of a function $y = f (x)$ at $x = a .$ Find $f (x)$ and $a .$
$lim_{h \to 0} \frac{[ 3 ( 2 + h ) ^{2} + 2 ] - 14}{h}$
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Exercises 71
For the following exercises, the given limit represents the derivative of a function $y = f (x)$ at $x = a .$ Find $f (x)$ and $a .$
$lim_{h \to 0} \frac{( 2 + h ) ^{4} - 16}{h}$
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Exercises 73
For the following exercises, the given limit represents the derivative of a function $y = f (x)$ at $x = a .$ Find $f (x)$ and $a .$
$lim_{h \to 0} \frac{e ^{h} - 1}{h}$
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Exercises 75
For the following functions,
sketch the graph and
use the definition of a derivative to show that the function is not differentiable at $x = 1.$
$f (x) = {3, x < 1 3 x, x \geq 1$
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Exercises 77
For the following functions,
sketch the graph and
use the definition of a derivative to show that the function is not differentiable at $x = 1.$
$f (x) = {2 x, x \leq 1 \frac{2}{x}, x > 1$
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Exercises 79
For the following graphs,
determine for which values of $x = a$ the $lim_{x \to a} f (x)$ exists but $f$ is not continuous at $x = a,$ and
determine for which values of $x = a$ the function is continuous but not differentiable at
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Exercises 81
For the following functions, use $f^{''} (x) = lim_{h \to 0} \frac{f ^{'} ( x + h ) - f ^{'} ( x )}{h}$ to find
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Exercises 83
For the following functions, use $f^{''} (x) = lim_{h \to 0} \frac{f ^{'} ( x + h ) - f ^{'} ( x )}{h}$ to find
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Exercises 85
For the following exercises, use a calculator to graph $f (x) .$ Determine the function $f^{'} (x),$ then use a calculator to graph $f^{'} (x) .$
[T] $f (x) = 3 x^{2} + 2 x + 4.$
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Exercises 87
For the following exercises, use a calculator to graph $f (x) .$ Determine the function $f^{'} (x),$ then use a calculator to graph $f^{'} (x) .$
[T] $f (x) = \frac{1}{2 x}$
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Exercises 89
For the following exercises, use a calculator to graph $f (x) .$ Determine the function $f^{'} (x),$ then use a calculator to graph $f^{'} (x) .$
[T] $f (x) = x^{3} + 1$
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Exercises 91
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
$\frac{f ( x + h ) - f ( x )}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
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Exercises 93
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
$\frac{f ( x + h ) - f ( x )}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
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Exercises 95
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
$\frac{f ( x + h ) - f ( x )}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
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Exercises 97
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
$\frac{f ( x + h ) - f ( x )}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
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Exercises 99
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
$\frac{f ( x + h ) - f ( x )}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
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Exercises 101
For the following exercises, use the following table, which shows the height $h$ of the Saturn $V$ rocket for the Apollo $11$ mission $t$ seconds after launch.
Time (seconds) Height (meters)
$0$ $0$
$1$ $2$
$2$
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Exercises 103
For the following exercises, use the following table, which shows the height $h$ of the Saturn $V$ rocket for the Apollo $11$ mission $t$ seconds after launch.
Time (seconds) Height (meters)
$0$ $0$
$1$ $2$
$2$
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Exercises 105
For the following exercises, use the following table, which shows the height $h$ of the Saturn $V$ rocket for the Apollo $11$ mission $t$ seconds after launch.
Time (seconds) Height (meters)
$0$ $0$
$1$ $2$
$2$
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Review Exercises 367
True or False? Justify the answer with a proof or a counterexample.
Every function has a derivative.
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Review Exercises 369
True or False? Justify the answer with a proof or a counterexample.
A continuous function has a derivative.
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Review Exercises 371
Use the limit definition of the derivative to exactly evaluate the derivative.
$f (x) = x + 4$
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Review Exercises 387
Draw the derivative for the following graphs.
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Key terms · 5

derivative function: gives the derivative of a function at each point in the domain of the original function for which the derivative is defined
differentiable at a: a function for which $f' (a)$ exists is differentiable at $a$
differentiable on S: a function for which $f' (x)$ exists for each $x$ in the open set $S$ is differentiable on $S$
differentiable function: a function for which $f' (x)$ exists is a differentiable function
higher-order derivative: a derivative of a derivative, from the second derivative to the nth derivative, is called a higher-order derivative

Learn 3.2 on the whiteboard

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

Read 3.2 in the textbook · This section on openstax.org

The Derivative as a Function: practice problems

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

Exercises 55