Chain Rule: the Derivative of a Composition

Description:
A lot of functions like sin(x^3) are the composition of an outside function with an inside function. The chain rule tells us how to compute the derivative of such functions.

Learning Objectives:
1) Compute the derivative of a composition.

Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

This video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

[robertolopez]

Perfectly explained. Here’s your like sir👍

[coleabrahams]

One thing I don’t get is the idea of derivatives of exponentials and the application of chain rule - I’ll give some substance to my confusion below:

I get that the chain rule can be applied to functions such as (3x+4)^5 because in a sense, the inner function is x^5 and the inner function is 3x+4 and thus, because one function is inside the other, it is a composite function and thus, the chain rule can be applied.

However, my confusion arises with exponentials and where the variable is placed as sometimes a particular rule such as the power rule is used and other times the chain rule is used. I.e. I get that d/dx (x^2) (where the variable is in the base) is 2x and this is due to the power rule and can also be seen through the definition of the derivative.
I’ve also accepted the rule that d/dx(a^x) (where the variable is the exponent) = (a^x )(ln(a)) -> I will look for the derivation of this soon. I understand that the derivative of e^x is e^x and in the same manner, the recently aforementioned rule with a^x can be applied. My confusion comes in when there is e.g. e^(x+2) - I see that it is a composite function where f(x) = e^x and g(x) = x+2 and thus, the chain rule can be applied... tbh I’m struggling to determine my exact confusion, but I seem to be baffled because every exponential function is a composite function, however, it’s not just the chain rule that is applied to all exponential functions, but rather different rules that must be applied to various types of exponential functions depending on its type.

Please help me, Professor or anyone else if you can

[vtrandal]

Excellent. At 1:00 into this video you express the composition as a sequence that uses cached results in each next step. For example you wrote x->x^3->sin(x^3). I love this because it seems to make an excellent foundation for explaining how neural networks are implemented. I'm making this observation from watching a other videos on this topic of neural networks. If I am on the right track then I expect a Deep Neural Network (with many hidden layers) can be expressed as a composition of functions. And when you unpack that composition as sequence of layers each used the result cached from the previous layer you have a framework upon which to implement "Automatic Differentiation." Am I on the right track? Am I understanding you?

[johnlouisville]

great explanation...thanks. Your accent is SOOO Canadian...