Chapter 4
Derivative Basics
T
he derivative is one of the great ideas in calculus. In this chapter, you see the formal
definition of a derivative. Understanding the formal definition is crucial, because it tells
you what a derivative actually is. Unfortunately, computing the derivative using the definition can be quite cumbersome and is often very difficult. After finding derivatives using the
definition, you see problems that use the power rule, which is the start of some techniques
that make finding the derivative much easier — although still challenging in many cases.
The Problems You’ll Work On
In this chapter, you see the definition of a derivative and one of the first shortcut formulas,
the power rule. Here’s what the problems cover:
✓ Using a variety of algebraic techniques to find the derivative using the definition of a
derivative
✓ Evaluating the derivative at a point using a graph and slopes of tangent lines
✓ Encountering a variety of derivative questions that you can solve using the power rule
What to Watch Out For
Using the definition of a derivative to evaluate derivatives can involve quite a bit of algebra,
so be prepared. Having all the shortcut techniques is very nice, but you’ll be asked to find
derivatives for complicated functions, so the problems will still be challenging! Keep some
of the following points in mind:
✓ Remember your algebra techniques: factoring, multiplying by conjugates, working with
fractions, and more. Many students get tripped up on one part and then can’t finish the
problem, so know that many problems require multiple steps.
✓ When interpreting the value of a derivative from a graph, think about the slope of the
tangent line on the graph at a given point; you’ll be well on your way to finding the
correct solution.
✓ Simplifying functions using algebra and trigonometric identities before finding the
derivative makes many problems much easier. Simplifying is one of the very first things
you should consider when encountering a “find the derivative” question of any type.
Derivative Basics
T
he derivative is one of the great ideas in calculus. In this chapter, you see the formal
definition of a derivative. Understanding the formal definition is crucial, because it tells
you what a derivative actually is. Unfortunately, computing the derivative using the definition can be quite cumbersome and is often very difficult. After finding derivatives using the
definition, you see problems that use the power rule, which is the start of some techniques
that make finding the derivative much easier — although still challenging in many cases.
The Problems You’ll Work On
In this chapter, you see the definition of a derivative and one of the first shortcut formulas,
the power rule. Here’s what the problems cover:
✓ Using a variety of algebraic techniques to find the derivative using the definition of a
derivative
✓ Evaluating the derivative at a point using a graph and slopes of tangent lines
✓ Encountering a variety of derivative questions that you can solve using the power rule
What to Watch Out For
Using the definition of a derivative to evaluate derivatives can involve quite a bit of algebra,
so be prepared. Having all the shortcut techniques is very nice, but you’ll be asked to find
derivatives for complicated functions, so the problems will still be challenging! Keep some
of the following points in mind:
✓ Remember your algebra techniques: factoring, multiplying by conjugates, working with
fractions, and more. Many students get tripped up on one part and then can’t finish the
problem, so know that many problems require multiple steps.
✓ When interpreting the value of a derivative from a graph, think about the slope of the
tangent line on the graph at a given point; you’ll be well on your way to finding the
correct solution.
✓ Simplifying functions using algebra and trigonometric identities before finding the
derivative makes many problems much easier. Simplifying is one of the very first things
you should consider when encountering a “find the derivative” question of any type.