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Unit 2 Overview

6 min readdecember 23, 2022

Sumi Vora

Sumi Vora

Sumi Vora

Sumi Vora

In this unit, you will learn more about derivatives, how they are defined, and the basic rules (and exceptions). As one of the core foundations of analyzing functions and change in calculus itself, it is of utmost importance to gain an in-depth understanding of this unit before moving forward to more advanced concepts.📑 This unit makes up 10-12% of the Calculus AB exam and 4-7% of the Calculus BC exam.

This list will be a good place to start in terms of self-assessing what you need to study or learn. 📖 It will also (hopefully) help you intuit a lot of the concepts that you will learn later in calculus. You can get into the nitty gritty later, but for now, try to focus on really understanding. As you are looking through this list, write down what topics you don’t remember or still need to learn.  You can then go to the full study guide page for each topic to get some more information!

What is a derivative?

Suppose we are watching Michael Phelps swim a 200m race, and on this particular day, he takes 1 minute and 40 seconds to complete it. If I asked you his average speed, you might use the formula speed = distance/time and say that his average speed was 200m/100s = 2 m/s. But, he didn't swim the whole race at exactly the same speed. Maybe I want to know which exact time in the race he was swimming the fastest, or maybe I want to know his speed at exactly the 1 minute mark. "But you can't find speed for just one instant in time! That doesn't make any sense!" you might argue. You would be correct. However, we have a little work-around to approximate this speed.

First, I want to define a real-valued function with time (t) on the horizontal axis and position (s) on the vertical axis. You could switch around the axes, but you'll see why making this particular choice makes sense in just a second. Now, the speed for some interval will just be the slope of the function in some particular interval, since slope = change in vertical axis / change in horizontal axis = change in distance / change in speed. Now, we need to figure out how to approximate the slope for one particular value of t. We can take a really small interval, say 0.1 seconds, and say that the speed can't change much within such a small time frame, so this is a pretty good approximation. Then, the slope would be given by:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.32-jzkwD2hnoTCq.png?alt=media&token=9d1ebad4-3bca-4188-9e6c-142eb9593832

But maybe an interval of 0.1 seconds isn't good enough for us. Maybe we want 0.01, or 0.00001 seconds. As the interval decreases towards 0, the closer we get to estimating the actual instantaneous speed. Let's use the variable dt to denote the size of the interval. Then, the instantaneous rate of speed is given by:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.32-Gy2IdYV0FKhP.png?alt=media&token=b8ad0423-f12d-425f-a1ad-39b87c1d1fbe

The of a function is called the . Usually, we say "the with respect to _". In this case, it will be "the with respect to t," since we are finding the rate of change of position with respect to time.

Notice that the deltas changed to ds in this equation. In math, whenever you see d's, we usually thing derivatives or .

I did this example using , because this is what we usually use when we're talking about velocity and speeds. But, you'll often find derivatives defined in the xy-plane, and instead of using small 's, we'll use h to denote the small change in x. For some real-valued function f on the xy-plane, the of f with respect to x is:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.33-MUQ3ztSaLYS8.png?alt=media&token=31115996-18bd-43d7-a1b8-c71f5d47fa6f

You will also see the notation f'(x) or y' (read f prime of x or y prime) for derivatives in two-dimensional spaces. This is okay and you should be familiar with the notation, but it is good to get into the habit of writing it in the form dy/ because this notation is used more in the real world (where there are more than two variables).

The derivative is the slope of the tangent line!

Given that the is the instantaneous slope of a function, the is also the slope of the tangent line of the function at that point. Make sure to remember this! There will be lots of applications later on.

A good way to intuit this idea is to imagine zooming in on the function, so you can approximate the slope of the function at some point as just a line. This is the tangent line. This line will just touch the function, For simplicity's sake, some books will say that the tangent line only touches the function at one point. However, this isn't always true, so don't stress if your tangent line crosses the function at another point.

An important part of the definition of a is that for some point, there will only exist one . That implies that there will only exist one tangent line. There are some functions, like piecewise functions and , where you could hypothetically draw two or three or an infinite number of tangent lines to the function. If that is the case, it is a good indication that the doesn't exist. Additionally, if you draw a tangent line and the line is exactly vertical, that means that the slope is infinity, which means that the is undefined. In general, a good heuristic is that if the function is , , and has a non-vertical slope, then the exists.

Here's an example of a function with a :

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.43-QN6thHgLw9DB.png?alt=media&token=2f9491ea-bb31-45fb-975d-3b31ea83871f

Computing Derivatives

Here are some good-to-know properties of derivatives. There are some good intuitive geometric explanations for why the sum, product, and quotient rules make sense, but it is usually easier to just use the formulas. However, don't spend too much time memorizing these. You can just do a few practice problems and get the hang of it.

  • : the of any constant is 0

  • : if you have a function with a constant multiplied by an expression, the of the function is just the of the expression times the constant. For example, d/(3x) = 3 * d/(x)

  • : if you have two expressions that are added together in a function, then the of the function is the sum of the derivatives of these expressions. For example, if you have f(x) = sin(x) + 3x, then f'(x) = d/(sin(x)) + d/(3x)

  • : The helps you find the of two expressions multiplied together. If h(x) = f(x) g(x), then h'(x) = f'(x)g(x) + f(x)g'(x)

  • : The comes from the . If you want to find the of h(x) = f(x)/g(x) = f(x) g⁻¹(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))²

There are several formulas available for computing derivatives. This can make finding derivatives very easy, but it somewhat obfuscates the definition and applications of derivatives. However, it is important to remember that these formulas are just derived from the definition of the . Here is a simple example:

Suppose you want to find the of the function f(x) = xⁿ. Then, we get:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.33-Z3LutcGdjhKz.png?alt=media&token=2db88c78-2453-46ed-9e72-4999acb223db

This is called the .

Special Derivatives

There are some special functions that we should just know the derivatives for. Here they are:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.33-ejlf4rKMEluz.png?alt=media&token=7e23bea0-ef29-4a22-af0d-31765117d27d

Example:

Find the of f(x) = x³ + 3x ln(x) + 5

Solution:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.34-FIs9alKULUzm.png?alt=media&token=45736631-76d6-46cc-9a92-939b199d666e

Key Terms to Review (16)

Absolute Value Functions

: Absolute value functions are functions that represent the distance between a number and zero on a number line. They can be written as f(x) = |x|, where f(x) is the output (y-value) and x is the input (x-value).

Constant Multiple Rule

: The constant multiple rule states that when taking the derivative of a function multiplied by a constant term (a number), you can simply multiply that constant term by the derivative of the original function.

Constant Rule

: The constant rule states that the derivative of a constant term (a number) is always zero. In other words, when you take the derivative of a constant, it disappears.

Continuous

: A function is continuous if there are no breaks, jumps, or holes in its graph. In other words, you can draw the graph of a continuous function without lifting your pencil.

Derivative

: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.

Differential Equations

: Differential equations are mathematical equations that involve derivatives. They describe how a function changes over time or in relation to other variables.

ds/dt

: ds/dt represents the derivative with respect to time. It measures how much one variable changes with respect to another variable (usually time).

dx

: In calculus, dx represents an infinitesimally small change or increment in x. It is often used when finding derivatives or integrating functions.

Instantaneous Rate of Change

: The instantaneous rate of change refers to the rate at which a function is changing at a specific point. It measures how quickly the output of a function is changing with respect to the input at that particular instant.

Power Rule

: The power rule is a calculus rule used to find the derivative of a function that is raised to a constant power. It states that if f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is equal to n*x^(n-1).

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Quotient Rule

: The quotient rule is a formula used to find the derivative of a quotient or division between two functions. It states that the derivative of a fraction is equal to (the denominator times the derivative of numerator) minus (the numerator times the derivative of denominator), all divided by (the square root of denominator squared).

Smooth

: In calculus, smooth refers to functions that are continuous and have no sharp corners or breaks in their graph. They exhibit gradual changes and have no abrupt changes in direction.

Sum Rule

: The sum rule is a calculus rule that states that the derivative of the sum of two functions is equal to the sum of their derivatives.

Vertical Tangent Line

: A vertical tangent line is a line that is perpendicular to the curve of a function at a specific point and has an undefined slope. It occurs when there is an abrupt change in direction or when the slope becomes infinite.

y'

: The derivative of a function with respect to the independent variable. It represents the rate of change of the function at any given point.

Unit 2 Overview

6 min readdecember 23, 2022

Sumi Vora

Sumi Vora

Sumi Vora

Sumi Vora

In this unit, you will learn more about derivatives, how they are defined, and the basic rules (and exceptions). As one of the core foundations of analyzing functions and change in calculus itself, it is of utmost importance to gain an in-depth understanding of this unit before moving forward to more advanced concepts.📑 This unit makes up 10-12% of the Calculus AB exam and 4-7% of the Calculus BC exam.

This list will be a good place to start in terms of self-assessing what you need to study or learn. 📖 It will also (hopefully) help you intuit a lot of the concepts that you will learn later in calculus. You can get into the nitty gritty later, but for now, try to focus on really understanding. As you are looking through this list, write down what topics you don’t remember or still need to learn.  You can then go to the full study guide page for each topic to get some more information!

What is a derivative?

Suppose we are watching Michael Phelps swim a 200m race, and on this particular day, he takes 1 minute and 40 seconds to complete it. If I asked you his average speed, you might use the formula speed = distance/time and say that his average speed was 200m/100s = 2 m/s. But, he didn't swim the whole race at exactly the same speed. Maybe I want to know which exact time in the race he was swimming the fastest, or maybe I want to know his speed at exactly the 1 minute mark. "But you can't find speed for just one instant in time! That doesn't make any sense!" you might argue. You would be correct. However, we have a little work-around to approximate this speed.

First, I want to define a real-valued function with time (t) on the horizontal axis and position (s) on the vertical axis. You could switch around the axes, but you'll see why making this particular choice makes sense in just a second. Now, the speed for some interval will just be the slope of the function in some particular interval, since slope = change in vertical axis / change in horizontal axis = change in distance / change in speed. Now, we need to figure out how to approximate the slope for one particular value of t. We can take a really small interval, say 0.1 seconds, and say that the speed can't change much within such a small time frame, so this is a pretty good approximation. Then, the slope would be given by:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.32-jzkwD2hnoTCq.png?alt=media&token=9d1ebad4-3bca-4188-9e6c-142eb9593832

But maybe an interval of 0.1 seconds isn't good enough for us. Maybe we want 0.01, or 0.00001 seconds. As the interval decreases towards 0, the closer we get to estimating the actual instantaneous speed. Let's use the variable dt to denote the size of the interval. Then, the instantaneous rate of speed is given by:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.32-Gy2IdYV0FKhP.png?alt=media&token=b8ad0423-f12d-425f-a1ad-39b87c1d1fbe

The of a function is called the . Usually, we say "the with respect to _". In this case, it will be "the with respect to t," since we are finding the rate of change of position with respect to time.

Notice that the deltas changed to ds in this equation. In math, whenever you see d's, we usually thing derivatives or .

I did this example using , because this is what we usually use when we're talking about velocity and speeds. But, you'll often find derivatives defined in the xy-plane, and instead of using small 's, we'll use h to denote the small change in x. For some real-valued function f on the xy-plane, the of f with respect to x is:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.33-MUQ3ztSaLYS8.png?alt=media&token=31115996-18bd-43d7-a1b8-c71f5d47fa6f

You will also see the notation f'(x) or y' (read f prime of x or y prime) for derivatives in two-dimensional spaces. This is okay and you should be familiar with the notation, but it is good to get into the habit of writing it in the form dy/ because this notation is used more in the real world (where there are more than two variables).

The derivative is the slope of the tangent line!

Given that the is the instantaneous slope of a function, the is also the slope of the tangent line of the function at that point. Make sure to remember this! There will be lots of applications later on.

A good way to intuit this idea is to imagine zooming in on the function, so you can approximate the slope of the function at some point as just a line. This is the tangent line. This line will just touch the function, For simplicity's sake, some books will say that the tangent line only touches the function at one point. However, this isn't always true, so don't stress if your tangent line crosses the function at another point.

An important part of the definition of a is that for some point, there will only exist one . That implies that there will only exist one tangent line. There are some functions, like piecewise functions and , where you could hypothetically draw two or three or an infinite number of tangent lines to the function. If that is the case, it is a good indication that the doesn't exist. Additionally, if you draw a tangent line and the line is exactly vertical, that means that the slope is infinity, which means that the is undefined. In general, a good heuristic is that if the function is , , and has a non-vertical slope, then the exists.

Here's an example of a function with a :

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.43-QN6thHgLw9DB.png?alt=media&token=2f9491ea-bb31-45fb-975d-3b31ea83871f

Computing Derivatives

Here are some good-to-know properties of derivatives. There are some good intuitive geometric explanations for why the sum, product, and quotient rules make sense, but it is usually easier to just use the formulas. However, don't spend too much time memorizing these. You can just do a few practice problems and get the hang of it.

  • : the of any constant is 0

  • : if you have a function with a constant multiplied by an expression, the of the function is just the of the expression times the constant. For example, d/(3x) = 3 * d/(x)

  • : if you have two expressions that are added together in a function, then the of the function is the sum of the derivatives of these expressions. For example, if you have f(x) = sin(x) + 3x, then f'(x) = d/(sin(x)) + d/(3x)

  • : The helps you find the of two expressions multiplied together. If h(x) = f(x) g(x), then h'(x) = f'(x)g(x) + f(x)g'(x)

  • : The comes from the . If you want to find the of h(x) = f(x)/g(x) = f(x) g⁻¹(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))²

There are several formulas available for computing derivatives. This can make finding derivatives very easy, but it somewhat obfuscates the definition and applications of derivatives. However, it is important to remember that these formulas are just derived from the definition of the . Here is a simple example:

Suppose you want to find the of the function f(x) = xⁿ. Then, we get:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.33-Z3LutcGdjhKz.png?alt=media&token=2db88c78-2453-46ed-9e72-4999acb223db

This is called the .

Special Derivatives

There are some special functions that we should just know the derivatives for. Here they are:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.33-ejlf4rKMEluz.png?alt=media&token=7e23bea0-ef29-4a22-af0d-31765117d27d

Example:

Find the of f(x) = x³ + 3x ln(x) + 5

Solution:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-23%2010.34-FIs9alKULUzm.png?alt=media&token=45736631-76d6-46cc-9a92-939b199d666e

Key Terms to Review (16)

Absolute Value Functions

: Absolute value functions are functions that represent the distance between a number and zero on a number line. They can be written as f(x) = |x|, where f(x) is the output (y-value) and x is the input (x-value).

Constant Multiple Rule

: The constant multiple rule states that when taking the derivative of a function multiplied by a constant term (a number), you can simply multiply that constant term by the derivative of the original function.

Constant Rule

: The constant rule states that the derivative of a constant term (a number) is always zero. In other words, when you take the derivative of a constant, it disappears.

Continuous

: A function is continuous if there are no breaks, jumps, or holes in its graph. In other words, you can draw the graph of a continuous function without lifting your pencil.

Derivative

: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.

Differential Equations

: Differential equations are mathematical equations that involve derivatives. They describe how a function changes over time or in relation to other variables.

ds/dt

: ds/dt represents the derivative with respect to time. It measures how much one variable changes with respect to another variable (usually time).

dx

: In calculus, dx represents an infinitesimally small change or increment in x. It is often used when finding derivatives or integrating functions.

Instantaneous Rate of Change

: The instantaneous rate of change refers to the rate at which a function is changing at a specific point. It measures how quickly the output of a function is changing with respect to the input at that particular instant.

Power Rule

: The power rule is a calculus rule used to find the derivative of a function that is raised to a constant power. It states that if f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is equal to n*x^(n-1).

Product Rule

: The product rule is a formula used to find the derivative of the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Quotient Rule

: The quotient rule is a formula used to find the derivative of a quotient or division between two functions. It states that the derivative of a fraction is equal to (the denominator times the derivative of numerator) minus (the numerator times the derivative of denominator), all divided by (the square root of denominator squared).

Smooth

: In calculus, smooth refers to functions that are continuous and have no sharp corners or breaks in their graph. They exhibit gradual changes and have no abrupt changes in direction.

Sum Rule

: The sum rule is a calculus rule that states that the derivative of the sum of two functions is equal to the sum of their derivatives.

Vertical Tangent Line

: A vertical tangent line is a line that is perpendicular to the curve of a function at a specific point and has an undefined slope. It occurs when there is an abrupt change in direction or when the slope becomes infinite.

y'

: The derivative of a function with respect to the independent variable. It represents the rate of change of the function at any given point.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.