First and foremost, you’ll be happy to know I survived the canker sore that just about killed me last week. It slowly healed throughout the week and was gone by Saturday, THANK GOD.

Overall, I’m pretty happy with the amount of work I got done this week although I probably could (and should) have gotten more done. Over the past 2-3 weeks, I think I’ve worked on KA for slightly more than 5 hours/week which is the minimum I set for myself but I’d like to be getting closer to 7.5 hours per week. Regardless, I did learn a handful of new things this week which I was happy about. For the most part, I haven’t found anything I’ve learned so far regarding derivatives too difficult to understand but some concepts still haven’t fully concretized in my head. I feel like I’m making progress understanding everything but it’s going slower than I’d like it to. I’m trying to remind myself that this is a marathon and not a sprint but it’s still a bit annoying.

The first thing I learned this week was the three different forms of derivative notation:

Sal said that each form of notation is used in different scenarios but that the most commonly used form was the first one int he above photo, Lagrange’s notation. This form was the only type of notation that came up in the videos I watched and the exercises I went through throughout the week. I can’t remember exactly, but I think Sal said the second form, Leibniz’s notation, is very useful when solving derivatives that are a bit more complicated than the ones I worked through this week. The last form, Newton’s notation, is apparently predominantly used in physics which I want to learn at some point so I was excited to see that what I’m learning now will eventually be used when I get to physics. 

What sunk in for me the most this week is that a derivative, f’(x), can be thought of as the slope, m, for a single point on a function:

In order to find the derivative of a point on a function, Sal went through a number of videos that talked about Secant lines on a function, Tangent lines, and the difference between the two:

A secant line is a line that connects two points on a function whereas a tangent line is a line that just ’grazes’ the function (so to speak) at one specific point and indicates the slope of the function at that specific point. One example that helps me visualize what a tangent line is would be thinking of a ball sitting on the ground with the ground being the ‘tangent line’ since it (theoretically) only touches the ball at one specific point.

To calculate the slope of a secant line, you use the standard “change-in-y­” over “change-in-x” formula, i.e. Δy/Δx = y₂ – y₁/x₂ – x₁, a.k.a. the ‘rise-over-run’. The key concept to understand when finding a derivative is that by bringing two points of a secant line closer and closer where the difference between the two points approaches 0, the secant line gets closer and closer to turning into a tangent line which is the slope of the function at that specific point which is the derivative of the function at that specific point. (That was a long sentence but I’m pretty sure it makes sense.)

The last thing I worked on this week was understanding the Formal Form and Alternate Form of a derivative:

• Formal Form
  • f'(x) = lim _h->0 ((f(x+h) – (f(x))/h)
    • “The derivative of a certain x value (a.k.a. f’(x)) equals the limit as h approaches 0 on the point f(x + h), minus f(x), divided by h.”
  • This formula states that the derivative of f(x) is calculated by by bringing point P₂ closer and closer to point P₁ (i.e. the distance between the two points, h, gets smaller and smaller and approaches 0) and then dividing the difference by h.
    
• Alternate Form
  • f'(x) = lim _b->a ((f(b) – (f(a))/b – a
    • “The derivative of a certain value a (a.k.a. f’(a)) equals the limit as b approaches a in the expression f(b) minus f(a), divided by b – a.”
  • This is saying the same thing as the formal form except instead of using h to describe the distance between two points getting smaller the alternate form states there are two points, a and b, which are getting closer and closer and the difference between the two points approaches 0.

I’m still having a hard time understanding the nuance between the formal and alternate forms of a derivative but I don’t think it will take me much longer to be able to recognize the differences and differentiate between the two forms.

I’m only 18% of the way through Derivatives: Definitions and Basic Rules (400/2500 M.P.) so I’m hoping this coming week I can get close to ~50% of the way done. To do that, I’ll have to get through 10 videos and 10 exercises (each exercise is worth 80 M.P. before the unit test). I think this is doable especially if I’m able to do ~7.5 hours in total and ESPECIALLY since I don’t have giant canker sore in my mouth anymore. 🥳🎊🎉