## Subtitles section Play video

• What we're gonna do in this video

• is focus on key misunderstandings that folks often have,

• and we actually got these misunderstandings

• from the folks who write the AP exams,

• from the actual College Board.

• So let's say that we are trying to take

• the derivative of the expression.

• So let's say we're taking the derivative

• of the expression,

• the natural log of sine of x.

• So the first key misconception or misunderstanding

• that many people have

• is when you're dealing

• with transcendental functions like this,

• and transcendental functions is just a fancy word

• for these functions like trigonometric functions,

• logarithmic functions,

• that don't use standard algebraic operations.

• But when you see transcendental functions like this

• or compositions of them,

• many people confuse this with the product of functions.

• So at first when they look at this,

• they might see this as being the same

• as the derivative with respect to x

• of natural log of x,

• natural log of x, times sine of x.

• And you can see just the way that it's written,

• they look very similar,

• but this is the product of two functions.

• If you said natural log of x is f of x,

• and sine of x is g of x,

• this is the product of sine and g of x,

• sorry this is the product of f of x and g of x,

• and here you would use the product rule.

• So to actually compute this,

• you would use the product, the product rule.

• But this is a composition.

• Here you have f of g of x,

• not f of x times g of x.

• So here you have

• that is our g of x, it equals sine of x,

• and then our f of g of x

• is the natural log of sine of x.

• So this is f of g of x,

• f of g of x just like that.

• If someone asks you just what f of x was,

• well that would be natural log of x,

• but f of g of x is natural log of our g of x,

• which is natural log of sine of x.

• So that's the key first thing,

• always make sure whether you're gonna use,

• especially with these transcendental functions,

• that hey if this is a composition you've gotta use

• the chain rule, not the product rule.

• It's not the product.

• Now sometimes you have a combination,

• you have a product of compositions,

• and then things get a little bit more involved.

• But pay close attention to make sure

• that you're not dealing with a composition.

• Now the next misconception students have

• is even if they recognize,

• okay I've gotta use the chain rule,

• sometimes it doesn't go fully to completion.

• So let's continue using this example.

• The chain rule here says,

• look we have to take the derivative of the outer function

• with respect to the inner function.

• So if I were to say,

• in this case, f of x is natural log of x,

• f of g of x is this expression here.

• So if I wanna do this first part,

• f prime of g of x,

• f prime of g of x,

• well the derivative of the natural log of x

• is one over x.

• So the natural log, derivative of natural log of x

• is one over x,

• but we don't want the derivative where the input is x.

• We want the derivative when the input is g of x.

• So instead of it being one over x,

• it's gonna be one over g of x.

• One over g of x,

• and we know that g of x is equal to sine of x.

• That's equal to sine of x.

• Now one key misunderstanding that the folks

• of the College Board told us about

• is many students stop right there.

• They just do this first part,

• and then they forget to multiply this second part.

• So here we are not done.

• We need to take this and multiply it times g prime of x.

• And let me write this down.

• g prime of x, what would that be?

• Well the derivative of sine of x with respect to x,

• well that's just going to be cosine of x,

• cosine of x.

• So in this example right over here,

• the derivative is going to be,

• let's see if I can squeeze it in over here,

• it's going to be one over sine of x which is this part,

• times cosine of x.

• So let me write it down.

• It is going to be one over sine of x,

• we'll do that in that other color,

• one over sine of x,

• and then times cosine of x.

• So once again,

• just to make sure that you don't fall into

• one of these misconceptions.

• Let me box this off so it's a little bit,

• it's a little bit cleaner.

• So to just make sure that you don't fall

• into one of these misconceptions here,

• recognize the composition,

• that this is not the product of natural log of x

• and sine of x.

• It's natural log of sine of x.

• And then when you're actually applying the chain rule,

• derivative of the outside with respect to the inside,

• so the derivative of natural log of x is one over x,

• so that applied when the input is g of x

• is one over sine of x.

• And then multiply that times the derivative

• of the inner function.

• So don't forget to do this right over here.

• Now another misconception that students have,

• is instead of doing what we just did,

• instead of applying the chain rule like this,

• they take the derivative of the outer function

• with respect to the derivative of the inner function.

• So for example,

• they would compute this,

• f prime of g prime of x,

• f prime of g prime of x.

• Which in this case, f prime of x is one over x,

• but if the input is g prime of x,

• g prime of x is cosine of x.

• So many students end up doing this

• where they take the derivative of the outside,

• and they apply the input into that,

• they use the derivative of the inside function.

• This is not right.

• Be very careful that you're not doing that.

• You do the derivative of the outside function

• with respect to the inside function,

• not taking it's derivative,

• and then multiply, don't forget to multiply,

• times the derivative of the inside function here.

• So hopefully that helps a little bit.

• If all of this looks completely foreign to you,

• I encourage you to watch the whole series

• of chain rule introductory videos

• and worked examples we have.

• This is just a topping on top of that

• to make sure that you don't fall into these misconceptions

• of applying the product rule

• when you really need to be applying the chain rule

• or forgetting to do part of the chain rule,

• multiplying by g prime of x,

• or evaluating f prime of g prime of x.

• So hopefully that helps.

What we're gonna do in this video

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# Common chain rule misunderstandings | Derivative rules | AP Calculus AB | Khan Academy

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yukang920108 posted on 2022/09/09
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