Can somebody explain the chain rule in an "easy-to-understand" way? Thanks :)

Consider a function (x+1)^2. We can see that the function has two parts, the enclosing part (outside: ( )^2) and the enclosed part (inside: x+1). To do the chain rule you first take the derivative of the outside as if you would normally (disregarding the inner parts), then you add the inside back into the derivative of the outside. Afterwards, you take the derivative of the inside part and multiply that with the part you found previously. So to continue the example: d/dx[(x+1)^2] 1. Find the derivative of the outside: Consider the outside ( )^2 as x^2 and find the derivative as d/dx x^2 = 2x the outside portion = 2( ) 2. Add the inside into the parenthesis: 2( ) = 2(x+1) 3. Find the derivative of the inside and multiply: as d/dx [x+1] = 1 1*2(x+1) = 2(x+1). Thus, d/dx[(x+1)^2] = 2(x+1)

In quotient rule, can't we write the derivative of f(x)/g(x) as f(x).g'(x)^-1 + g(x)^-1.f'(x) ?

No, because d/dx (1/g(x)) is not 1/g'(x). You have to apply the chain rule. It should be -1/(g(x))² · g'(x) So we'd have (using the product rule and the chain rule): d/dx f(x) · 1/g(x) = 1/g(x) · f'(x) + f(x) · -1/(g(x))² · g'(x) Which, with a bit of manipulation, can be made to look like the familiar quotient rule for differentiation.

In terms of the AP exam: Are proofs even needed? I know that you need to get a good conceptual idea, but my math teacher claims no one ever uses proofs. If they want, they can take a course on proofs in college, but teachers would lose kids interest with all these proofs. I'm 99 percent sure proofs are not required on the AP exam? Someone correct me if I'm wrong.

Proofs are NOT specifically needed for the AP exam. However, working through proofs can be illuminating into how functions (and their derivations) work. Practically speaking, the more you are able to manipulate functions algebraically and trigonometrically, the better you will be able to work with function problems at the AP (or any) exam. It's a matter of familiarity, fluency, and functionality. You don't have to be able to do a proof at the exam, but if you've done a lot of proofs beforehand, you'll likely score better at the exam.

Main content

AP®︎ Calculus AB (2017 edition)

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

Lesson 12: Strategy in differentiating functions

Derivative rules review

AP.CALC:

FUN‑3 (EU)

FUN‑3.C (LO)

FUN‑3.C.1 (EK)

Google Classroom

Review all the common derivative rules (including Power, Product, and Chain rules).

Basic differentiation rules

Constant rule: start fraction, d, divided by, d, x, end fraction, left parenthesis, k, right parenthesis, equals, 0

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, f, prime, left parenthesis, x, right parenthesis, plus, g, prime, 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, f, prime, left parenthesis, x, right parenthesis, minus, g, prime, left parenthesis, x, right parenthesis

Constant multiple rule: start fraction, d, divided by, d, x, end fraction, open bracket, k, f, left parenthesis, x, right parenthesis, close bracket, equals, k, f, prime, left parenthesis, x, right parenthesis

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

Power rule

start fraction, d, divided by, d, x, end fraction, left parenthesis, x, start superscript, n, end superscript, right parenthesis, equals, n, dot, x, start superscript, n, minus, 1, end superscript

Want to learn more about the Power rule? Check out this video.

Product rule

start fraction, d, divided by, d, x, end fraction, open bracket, f, left parenthesis, x, right parenthesis, g, left parenthesis, x, right parenthesis, close bracket, equals, f, prime, left parenthesis, x, right parenthesis, g, left parenthesis, x, right parenthesis, plus, f, left parenthesis, x, right parenthesis, g, prime, left parenthesis, x, right parenthesis

Want to learn more about the Product rule? Check out this video.

Quotient rule

start fraction, d, divided by, d, x, end fraction, left parenthesis, start fraction, f, left parenthesis, x, right parenthesis, divided by, g, left parenthesis, x, right parenthesis, end fraction, right parenthesis, equals, start fraction, f, prime, left parenthesis, x, right parenthesis, g, left parenthesis, x, right parenthesis, minus, f, left parenthesis, x, right parenthesis, g, prime, left parenthesis, x, right parenthesis, divided by, open bracket, g, left parenthesis, x, right parenthesis, close bracket, squared, end fraction

Want to learn more about the Quotient rule? Check out this video.

Chain rule

start fraction, d, divided by, d, x, end fraction, open bracket, f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, close bracket, equals, f, prime, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, g, prime, left parenthesis, x, right parenthesis

Want to learn more about the Chain rule? Check out this video.

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Moksha
Posted 3 years ago. Direct link to Moksha's post “Can somebody explain the ...”
Can somebody explain the chain rule in an "easy-to-understand" way? Thanks :)
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(10 votes)
Answer
- Seongjoo
  Posted 3 years ago. Direct link to Seongjoo's post “Consider a function (x+1)...”
  Consider a function (x+1)^2.
  We can see that the function has two parts, the enclosing part
  (outside: ( )^2) and the enclosed part (inside: x+1).
  
  To do the chain rule you first take the derivative of the outside as if you would normally (disregarding the inner parts), then you add the inside back into the derivative of the outside.
  Afterwards, you take the derivative of the inside part and multiply that with the part you found previously.
  
  So to continue the example:
  d/dx[(x+1)^2]
  1. Find the derivative of the outside:
  Consider the outside ( )^2 as x^2 and find the derivative
  as d/dx x^2 = 2x
  the outside portion = 2( )
  2. Add the inside into the parenthesis:
  2( ) = 2(x+1)
  3. Find the derivative of the inside and multiply:
  as d/dx [x+1] = 1
  1*2(x+1) = 2(x+1).
  
  Thus, d/dx[(x+1)^2] = 2(x+1)
  Button navigates to signup page
  (24 votes)
Anurag
Posted 6 years ago. Direct link to Anurag's post “In quotient rule, can't w...”
In quotient rule, can't we write the derivative of f(x)/g(x) as
f(x).g'(x)^-1 + g(x)^-1.f'(x) ?
Button navigates to signup pageComment on Anurag's post “In quotient rule, can't w...”
(2 votes)
Answer
- Howard Bradley
  Posted 6 years ago. Direct link to Howard Bradley's post “No, because d/dx (1/g(x))...”
  No, because d/dx (1/g(x)) is not 1/g'(x). You have to apply the chain rule. It should be -1/(g(x))² · g'(x)
  So we'd have (using the product rule and the chain rule):
  d/dx f(x) · 1/g(x) = 1/g(x) · f'(x) + f(x) · -1/(g(x))² · g'(x)
  
  Which, with a bit of manipulation, can be made to look like the familiar quotient rule for differentiation.
  Button navigates to signup page
  (9 votes)
abu
Posted 5 years ago. Direct link to abu's post “In terms of the AP exam: ...”
In terms of the AP exam: Are proofs even needed? I know that you need to get a good conceptual idea, but my math teacher claims no one ever uses proofs. If they want, they can take a course on proofs in college, but teachers would lose kids interest with all these proofs. I'm 99 percent sure proofs are not required on the AP exam? Someone correct me if I'm wrong.
Button navigates to signup pageComment on abu's post “In terms of the AP exam: ...”
(0 votes)
Answer
- Mark Geary
  Posted 5 years ago. Direct link to Mark Geary's post “Proofs are NOT specifical...”
  Proofs are NOT specifically needed for the AP exam. However, working through proofs can be illuminating into how functions (and their derivations) work. Practically speaking, the more you are able to manipulate functions algebraically and trigonometrically, the better you will be able to work with function problems at the AP (or any) exam. It's a matter of familiarity, fluency, and functionality. You don't have to be able to do a proof at the exam, but if you've done a lot of proofs beforehand, you'll likely score better at the exam.
  Button navigates to signup page
  (13 votes)
Melia Jane
Posted 3 years ago. Direct link to Melia Jane's post “Hey Great Learners. x(x-4...”
Hey Great Learners. x(x-4)^3 becomes (x-4)^2(4x-4) How is this when using the chain rule? I get 3x(x-4)^2(1).
Please and Thanks!
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(2 votes)
Answer
- Chace Caven
  Posted 3 years ago. Direct link to Chace Caven's post “For the equation `y = x(x...”
  For the equation y = x(x - 4) ^ 3 you have to use the product rule AND the chain rule since it's x * (x-4)^3. If you're using the formula d/dx f(x)g(x) = f'(x)g(x) + g'(x)f(x) then your f(x)=x and g(x)=(x-4)^3 so f'(x)=1 and g'(x)=3(x-4)^2, so if you plug it in you get 1 * (x-4)^3 + 3(x-4)^2 * x which I simplified to (x - 4)^2 (x-4+3x) or (x-4)^2(4x-4).
  
  Hope this help!
  Comment on Chace Caven's post “For the equation `y = x(x...”
  (1 vote)
James
Posted 5 days ago. Direct link to James's post “This has gotta be all for...”
This has gotta be all for Calculus AB, isn't it?
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(1 vote)
Answer
- Venkata
  Posted 5 days ago. Direct link to Venkata's post “Yes. Everything until the...”
  Yes. Everything until the application of integrals is AB Calc. Parametric equations and infinite sums are BC Calc
  Button navigates to signup page
  (1 vote)