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## AP®︎ Calculus AB (2017 edition)

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

Lesson 1: Basic differentiation rules- Proof of the constant derivative rule
- Proofs of the constant multiple and sum/difference derivative rules
- Basic derivative rules: find the error
- Basic derivative rules: find the error
- Basic derivative rules: table
- Basic derivative rules: table
- Basic differentiation review

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# Basic differentiation review

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;

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:

## Want to join the conversation?

- what are some easy ways of remembering differentiation and integration rules for o'level students(0 votes)
- 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.(13 votes)

- 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.(3 votes)
- Yes it is! Finding structure in these properties can only help you as you move through the course :)(2 votes)

- How can I discriminate all these different rule?(1 vote)
- 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!(5 votes)

- 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.)(1 vote)
- 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.(3 votes)

- 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?(1 vote)
- 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.(1 vote)

- do you find a derivittve of a function(1 vote)
- 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.(1 vote)

- f(y)=y/(y+x)..how do i find the derivative using the first principle?(0 votes)
- Actually, that equation f(y) = y/(y+x) is not solved by the principle. To do the derivative, more information has to be given.(1 vote)

- What is the differential coefficient of x^2/x^3+1?(0 votes)
- 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)(0 votes)