AP®︎/College Calculus AB
- Worked example: Chain rule with table
- Chain rule with tables
- Derivative of aˣ (for any positive base a)
- Derivative of logₐx (for any positive base a≠1)
- Derivatives of aˣ and logₐx
- Worked example: Derivative of 7^(x²-x) using the chain rule
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Worked example: Derivative of sec(3π/2-x) using the chain rule
- Worked example: Derivative of ∜(x³+4x²+7) using the chain rule
- Chain rule capstone
- Proving the chain rule
- Derivative rules review
Given the values of f and g (and their derivatives) at some x-values, Sal evaluates the derivative of F(x)=f(g(x)) at a specific x-value..
Want to join the conversation?
- is there anybody who can prove this chain rule?(3 votes)
- Sal has a great explanation for it later in the course:
- Given the chart, how would you know which part (top or bottom) applies to f and g?(2 votes)
- Answer is 40, cuz f'(-2)=5 ... isnt?(1 vote)
- No, we are trying to use the Chain Rule here.
d/dx f(g(x)) = f'(g(x))g'(x)
when x = 4, g(4) = -2
when x = -2, f'(-2) = 1
when x = 4, g'(4) = 8
1 * 8 = 8
- What does f-prime and g-prime mean?(0 votes)
- I need a proof for this magical rule(1 vote)
- Why do you have to multiply by g(x) in the chain rule?
Why is it not just f'(g'(x))?(1 vote)
- Are you aware of Leibnitz notation?(You know, the d/dx thing)
An intuition of the chain rule is that for an f(g(x)),
df/dx =df/dg * dg/dx.
If you look at this carefully, this is the chain rule.(1 vote)
- [Voiceover] The following table lists the values of functions f and g and of their derivatives, f-prime and g-prime for the x values negative two and four. And so you can see for x equals negative two, x equals four, they gave us the values of f, g, f-prime, and g-prime. Let function capital-F be defined as the composition of f and g. It's lowercase-f of g of x, and they want us to evaluate f-prime of four. So you might immediately recognize that if I have a function that can be viewed as the composition of other functions that the chain rule will apply here. And so, and I'm just gonna restate the chain rule, the derivative of capital-F is going to be the derivative of lowercase-f, the outside function with respect to the inside function. So lowercase-F-prime of g of x times the derivative of the inside function with respect to x times g-prime of x. And if we're looking for F-prime of four, F-prime of four, well everywhere we see an x we replace it with a four. That's gonna be lowercase-f-prime of g of four times g-prime of four. Now how do we figure this out? They haven't given us explicitly the values of the functions for all xs, but they've given it to us at some interesting points. So the first thing you might wanna figure out is well what is g of four going to be? Well they tell us: when x is equal to four, g of four is negative two. This tells us that the value of g of x takes on when x is equal to four is negative two. So this right over here is negative two. And so this first part is f-prime of negative two. So what is f-prime, what is f-prime of negative two? Well when x is equal to negative two, f-prime is equal to one. So this right over here is f-prime of negative two. That is equal to one. And now we just have to figure out what g-prime of four is. Well when, let me circle this, g-prime of four, when x is equal to four, and I'll scroll down a little bit, when x is equal to four, g-prime takes on the value eight. So there you have it. F-prime of four is equal to one times eight which is equal to eight, and we're done.