Warm up was to write the equation of a tangent line, where the derivative required the product rule. Reviewed what function types we are able to take derivatives of and completed the 'trig set' by finding the derivative of secant (and similarly, cosecant). Looked at a composition of functions and noted that its derivative is more nuanced than just the derivative of each part. With a simpler composition of functions (involving a quadratic) we were able to 'reverse engineer' the derivative's relationship to the original function, which led to the recognition of the chain rule pattern. This allows us to find derivatives of functions consisting of functions nested in other functions. Worked through several chain rule examples. Note that the chain rule can be used in tandem with the product and quotient rules as well as the power rule and trig derivative rules. It is a very important concept to understand in calculus.
Notes from board
Homework
all of this handout (due Monday) link
Resources
simple chain rule examples: link
more chain rule examples: link
proof of chain rule, for those interested: link
making a function differentiable (helps with first problem) link
Talked about differentiability in DS. A function is differentiable at a point if it is continuous there AND if there is a unique tangent line with a defined slope there. There are basically three criteria then: continuity, no ambiguity in the slope there, and the tangent line is not vertical. Moving from point to interval, a function is differentiable on an entire interval if it is differentiable at every point in the interval (kind of a no brainer). How this translates to functions as a whole is this: a function F is differentiable if it is continuous AND if its derivative F-Prime is also continuous. This is because the slope of F (which means the values of F-Prime) don't jump, have holes, or go to infinity. This guarantees smoothness. We use this fact about F and F-prime both being continuous to establish the parameters that make a function differentiable.
Notes from board
Video example similar to one done in class
Warm up was two derivative problems involving trig derivatives and the chain rule (as well as the product rule in one of them). We then looked back at our personal history with functions and noticed that we can take derivatives of almost all of them except for exponential and logarithmic functions. We reviewed some basic rules of exponents and logs as well as the definition of the natural number e, which was to prepare for proving the derivative of ln(x) using the limit definition. Through a lengthy proof, we saw that the derivative of ln(x) is 1/x. We used this fact to show that the derivative of e^x is itself: e^x. This means that the slope of the standard exponential growth curve is equivalent to its value. We worked through a few examples involving logs and exponentials as well as the chain rule.
Notes from board
Homework
all of this handout (due Monday) link
Resources
see Monday above for handout help
Log and Exponential Resources
video review of the proofs we did: link
some examples (made by me!) link
product and chain rule, combined with exp derivative (made by me!) first half of this
more examples link1 link2 link3 link4