Isn't the difference rule exactly the same as the sum rule? f(x) - g(x) is the same thing as f(x) + (-g(x)), after all.

Yes it is! Finding structure in these properties can only help you as you move through the course :)

How can I discriminate all these different rule?

Practice, practice, practice. You can go back and watch the videos again, and re-work the exercises and quizzes until you feel comfortable with them. Best wishes!

I am wondering if there are any prerequisites that I should be doing before I start AP Calculus. I am a fast learner and good at picking up new topics, but I am wondering if there are any holes in my education so far. (I am not doing this with school.)

You really need a good foundation in working with functions. You should be able to work with functions defined many different ways (tables, graphs, equations ect) and be able to recognize parent functions and apply transformations. You should also be able to use function notation very consistently with functions presented in any form. I have found that working hard with the students on this in their pre-calc courses made a huge difference to their success in AP calc.

what are some easy ways of remembering differentiation and integration rules for o'level students

My Calculus professor in college said there is only one way you can actually memorize most of the rules and learn to quickly and effectively apply them: Do a ton of exercises. There's no two ways about it, I'm afraid.

Main content

AP®︎ Calculus AB (2017 edition)

Course: AP®︎ Calculus AB (2017 edition) > Unit 3

Lesson 1: Basic differentiation rules

Basic differentiation review

Google Classroom

Review the basic differentiation rules and use them to solve problems.

What are the basic differentiation rules?


Sum rule	start fraction, d, divided by, d, x, end fraction, open bracket, f, left parenthesis, x, right parenthesis, plus, g, left parenthesis, x, right parenthesis, close bracket, equals, start fraction, d, divided by, d, x, end fraction, f, left parenthesis, x, right parenthesis, plus, start fraction, d, divided by, d, x, end fraction, g, left parenthesis, x, right parenthesis
Difference rule	start fraction, d, divided by, d, x, end fraction, open bracket, f, left parenthesis, x, right parenthesis, minus, g, left parenthesis, x, right parenthesis, close bracket, equals, start fraction, d, divided by, d, x, end fraction, f, left parenthesis, x, right parenthesis, minus, start fraction, d, divided by, d, x, end fraction, g, left parenthesis, x, right parenthesis
Constant multiple rule	start fraction, d, divided by, d, x, end fraction, open bracket, k, dot, f, left parenthesis, x, right parenthesis, close bracket, equals, k, dot, start fraction, d, divided by, d, x, end fraction, f, left parenthesis, x, right parenthesis
Constant rule	start fraction, d, divided by, d, x, end fraction, k, equals, 0

The Sum rule says the derivative of a sum of functions is the sum of their derivatives.

The Difference rule says the derivative of a difference of functions is the difference of their derivatives.

The Constant multiple rule says the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

The Constant rule says the derivative of any constant function is always 0.

Want to learn more about basic differentiation rules? Check out this video.

What problems can I solve with basic differentiation rules?

You can find the derivatives of functions that are combinations of other, simpler, functions. For example, H, left parenthesis, x, right parenthesis is defined as 2, f, left parenthesis, x, right parenthesis, minus, 3, g, left parenthesis, x, right parenthesis, plus, 5. We can find H, prime, left parenthesis, x, right parenthesis as follows;

\begin{aligned} &\phantom{=}H'(x) \\\\ &=\dfrac{d}{dx}H(x)&&\gray{\text{Equivalent notation}} \\\\ &=\dfrac{d}{dx}[2f(x)-3g(x)+5]&&\gray{\text{Substitute the expression for }H(x)} \\\\ &=\dfrac{d}{dx}[2f(x)]-\dfrac{d}{dx}[3g(x)]+\dfrac{d}{dx}(5)&&\gray{\text{Sum and difference rules}} \\\\ &=2f'(x)-3g'(x)+0&&\gray{\text{Constant and constant multiple rules}} \end{aligned}

We used the basic differentiation rules to find that H, prime, left parenthesis, x, right parenthesis, equals, 2, f, prime, left parenthesis, x, right parenthesis, minus, 3, g, prime, left parenthesis, x, right parenthesis.

Now suppose we are also given that start color #11accd, f, prime, left parenthesis, 3, right parenthesis, equals, 1, end color #11accd and start color #e07d10, g, prime, left parenthesis, 3, right parenthesis, equals, 5, end color #e07d10. We can find H, prime, left parenthesis, 3, right parenthesis as follows:

\begin{aligned} H'(3)&=2\blueD{f'(3)}-3\goldD{g'(3)} \\\\ &=2(\blueD1)-3(\goldD5) \\\\ &=-13 \end{aligned}

Check your understanding

Problem 1

Current

x	space, f, left parenthesis, x, right parenthesis	space, h, left parenthesis, x, right parenthesis	space, f, prime, left parenthesis, x, right parenthesis	space, h, prime, left parenthesis, x, right parenthesis
1	minus, 1	minus, 18	0	4

G, left parenthesis, x, right parenthesis, equals, minus, 4, f, left parenthesis, x, right parenthesis, plus, 3, h, left parenthesis, x, right parenthesis, minus, 2

G, prime, left parenthesis, 1, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

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prottoyee2001
Posted 6 years ago. Direct link to prottoyee2001's post “what are some easy ways o...”
what are some easy ways of remembering differentiation and integration rules for o'level students
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- Cristiano Morgado
  Posted 6 years ago. Direct link to Cristiano Morgado's post “My Calculus professor in ...”
  My Calculus professor in college said there is only one way you can actually memorize most of the rules and learn to quickly and effectively apply them: Do a ton of exercises. There's no two ways about it, I'm afraid.
  Comment on Cristiano Morgado's post “My Calculus professor in ...”
  (13 votes)
BB8FN2187
Posted 5 years ago. Direct link to BB8FN2187's post “Isn't the difference rule...”
Isn't the difference rule exactly the same as the sum rule? f(x) - g(x) is the same thing as f(x) + (-g(x)), after all.
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(3 votes)
Answer
- Ms Demaray
  Posted 5 years ago. Direct link to Ms Demaray's post “Yes it is! Finding struct...”
  Yes it is! Finding structure in these properties can only help you as you move through the course :)
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  (2 votes)
Christine Fan
Posted 5 years ago. Direct link to Christine Fan's post “How can I discriminate al...”
How can I discriminate all these different rule?
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(1 vote)
Answer
- Mark Geary
  Posted 5 years ago. Direct link to Mark Geary's post “Practice, practice, pract...”
  Practice, practice, practice. You can go back and watch the videos again, and re-work the exercises and quizzes until you feel comfortable with them. Best wishes!
  Button navigates to signup page
  (5 votes)
Robert
Posted 6 years ago. Direct link to Robert's post “I am wondering if there a...”
I am wondering if there are any prerequisites that I should be doing before I start AP Calculus. I am a fast learner and good at picking up new topics, but I am wondering if there are any holes in my education so far. (I am not doing this with school.)
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(1 vote)
Answer
- ketriwilkes
  Posted 6 years ago. Direct link to ketriwilkes's post “You really need a good fo...”
  You really need a good foundation in working with functions. You should be able to work with functions defined many different ways (tables, graphs, equations ect) and be able to recognize parent functions and apply transformations. You should also be able to use function notation very consistently with functions presented in any form. I have found that working hard with the students on this in their pre-calc courses made a huge difference to their success in AP calc.
  Button navigates to signup page
  (3 votes)
Aditya Shelke
Posted 6 years ago. Direct link to Aditya Shelke's post “On a graph assume y = x^2...”
On a graph assume y = x^2.If we know how y is going to change with respect to x.Then,why do we need to differentiate?
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(1 vote)
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- BB8FN2187
  Posted 5 years ago. Direct link to BB8FN2187's post “Just because we know a fu...”
  Just because we know a function is changing with respect to x isn't enough. In most cases we like to know how it is changing.
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  (1 vote)
610781
Posted 5 years ago. Direct link to 610781's post “do you find a derivittve ...”
do you find a derivittve of a function
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(1 vote)
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- mhyd1427
  Posted 10 months ago. Direct link to mhyd1427's post “Yes, you do need to find ...”
  Yes, you do need to find the derivative of the function that you're asked to find the derivative of! You can find the derivative of a function by applying the differentiation rules listed above.
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  (1 vote)
loboapiak
Posted 6 years ago. Direct link to loboapiak's post “f(y)=y/(y+x)..how do i fi...”
f(y)=y/(y+x)..how do i find the derivative using the first principle?
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- Yashwi P
  Posted 6 years ago. Direct link to Yashwi P's post “Actually, that equation f...”
  Actually, that equation f(y) = y/(y+x) is not solved by the principle. To do the derivative, more information has to be given.
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  (1 vote)
Aamir Naushad
Posted 5 years ago. Direct link to Aamir Naushad's post “What is the differential ...”
What is the differential coefficient of x^2/x^3+1?
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- Lexi Adams
  Posted 5 years ago. Direct link to Lexi Adams's post “The power rule will help ...”
  The power rule will help you with that, and so will the quotient rule. The former states that d/dx x^n = n*x^n-1, and the latter states that when you have a function such as the one you have described, the answer would be the derivative of x^2 multiplied by x^3 + 1, then you subtract x^2 multiplied by the derivative of x^3 - 1, and then divide all that by (x^3 - 1)^2. Basically, the answer would be (-x^4+2x)/(x^6+2x^3+1)
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