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## AP®︎ Calculus AB (2017 edition)

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

Lesson 1: Basic differentiation rules

# Basic differentiation review

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

## What are the basic differentiation rules?

Sum rulestart 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 rulestart 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 rulestart 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 rulestart 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.

## 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}

Problem 1
• Current
xspace, f, left parenthesis, x, right parenthesisspace, h, left parenthesis, x, right parenthesisspace, f, prime, left parenthesis, x, right parenthesisspace, h, prime, left parenthesis, x, right parenthesis
1minus, 1minus, 1804
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.

## Want to join the conversation?

• 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.
• 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?
(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!
• 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.
• 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?