2-the-chain-rule-how-mathbff

MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. More videos with Nancy coming in 2017! To skip ahead: 1) For how to use the CHAIN RULE or "OUTSIDE-INSIDE rule", skip to time 0:17. 1b) For how to know WHEN YOU NEED the chain rule, skip to time 4:35. 2) For another example with the POWER RULE in the chain rule, skip to time 7:05. 3) For a TRIG derivative chain rule example, skip to time 9:33. 3b) For the formal chain rule FORMULA, skip to time 11:36. PS) For a DOUBLE CHAIN RULE (or "repeated use of the chain rule") example, skip to time 13:33.

1) CHAIN RULE: You need the chain rule to take the derivative when you have a function inside a function, or a "composite function". For example, in the equation y = (3x + 1)^7, since the function 3x+1 is inside a larger, outer function, the power of 7, you will need the chain rule to find the correct derivative. How do you use the chain rule? You can think of the chain rule as the "OUTSIDE-INSIDE" rule: take the DERIVATIVE of JUST the OUTSIDE function first, LEAVING THE INSIDE FUNCTION alone (unchanged), and then MULTIPLY BY the DERIVATIVE of JUST the INSIDE function. Sometimes you might hear this expressed as: take the derivative of the outer function, "evaluated at the inner function", times the derivative of just the inner function. For our example, first take the derivative of the outer function (the power of 7) to get 7*(3x + 1)^6 since the power rule tells you that to take the derivative of a power you bring down the power to the front (as a constant or coefficient just multiplied in the front) and then you decrease the power by 1, which leaves a power of 6. Notice that you leave the inside function the way it is and just rewrite it for now. Then you multiply by the derivative of just the inner function, 3x + 1. Since the derivative of 3x + 1 is just 3, the full derivative (dy/dx) is: 7*[(3x + 1)^6]*3, which is just 21(3x + 1)^6.

1b) HOW do you know WHEN TO USE the chain rule? If the original equation had just been x^7, there would be no need for the chain rule. It's when you have something more than just x inside that you should use the chain rule, such as (3x + 1)^7 or even (x^2 + 1)^7. Sometimes the chain rule may make no difference. For instance, if you have the function (x + 1)^7, taking the derivative of the inside function just gives you 1, so multiplying by that inside derivative of 1 will not change the overall answer. However, it can't hurt to use the chain rule anyway, so it's a good idea to get in the habit of using it so that you don't forget it when it really does make a difference.

2) Another chain POWER RULE example: To find the derivative of h(x) = (x^2 + 5x - 6)^9, use the same steps as above to first take the outside derivative and then multiply by the inside derivative. In this case, the derivative, dh/dx (or h'(x)) is equal to 9(x^2 + 5x - 6)^8 * (2x + 5). Using the chain rule with the power rule is sometimes called the "power chain rule".

3) TRIG EXAMPLE: the idea is the same as above even if you are using the chain rule to differentiate something like a trigonometric function. If you have anything more than just x inside the trig function, you will need the chain rule to find the derivative. For the equation y = sin(x^2 - 3x), you first take the derivative of the outer function, just the sine function. Since the derivative of sine is cosine, the outside derivative (with the inside left unchanged) is cos(x^2 - 3x). Then, find the derivative of just the inside (of just the x^2 - 3x part), and multiply by that. Since the derivative of x^2 - 3x is 2x - 3, the full derivative answer is dy/dx = cos(x^2 - 3x)*(2x - 3).

3b) FORMULA: Although it's easier to think about the chain rule as the "outside-inside rule", if for any reason you have to use the formal chain rule formula, check out the two versions I show here. Both are based on the equation being a composition of functions, f(g(x)). The second version shown uses Liebniz notation. Either way, both show a component of the derivative that comes from the inside function, and it's important not to forget to multiply by this inside derivative factor if you want to get the right full derivative answer.

P.S.) DOUBLE CHAIN RULE: Sometimes you might have to use the chain rule more than once, known as "repeated use of the chain rule". In y = (1 + cos2x)^2, not only would you need to take the derivative of the outside power of 2, as well as multiply by the derivative of the inside function, 1 + cos2x, but after that you would ALSO then need to multiply by the derivative of the 2x inside cosine because that inside function was 1 + cos2x and not just 1 + cosx. Anyway, this means you would use the chain rule twice. The idea is that you have to keep taking the derivative of the inner functions until you have reached every inner function that is more complicated than just "x".

Tagged under: chain rule,derivative,calculus,composite function,dy/dx, ,differentiate,find derivative,formula,double chain rule,trig,trigonometric,differentiation,power chain rule,introduction,khan,patrickjmt,power rule,product rule,quotient rule,nested,linked,repeated,polynomial,logarithm,logs,liebniz,notation,algebra 2,natural log,ln ,exponent,exponential,simplify,square root,rule, ,solve,function,-,tutorial,explain,find,problem,math

Clip makes it super easy to turn any public video into a formative assessment activity in your classroom.

Add multiple choice quizzes, questions and browse hundreds of approved, video lesson ideas for Clip

Make YouTube one of your teaching aids - Works perfectly with lesson micro-teaching plans

1. Students enter a simple code

2. You play the video

3. The students comment

4. You review and reflect

* Whiteboard required for teacher-paced activities

With four apps, each designed around existing classroom activities, Spiral gives you the power to do formative assessment with anything you teach.

Quickfire

Carry out a quickfire formative assessment to see what the whole class is thinking

Discuss

Create interactive presentations to spark creativity in class

Team Up

Student teams can create and share collaborative presentations from linked devices

Clip

Turn any public video into a live chat with questions and quizzes

Tried out the canvas response option on @SpiralEducation & it's so awesome! Add text or drawings AND annotate an image! #R10tech

Using @SpiralEducation in class for math review. Student approved! Thumbs up! Thanks.

Absolutely amazing collaboration from year 10 today. 100% engagement and constant smiles from all #lovetsla #spiral

Students show better Interpersonal Writing skills than Speaking via @SpiralEducation Great #data #langchat folks!

Dr Ayla Göl
@iladylayla

A good tool for supporting active #learning.

The Team Up app is unlike anything I have ever seen. You left NOTHING out! So impressed!