One again, I was very happy with the amount of work I did this week but I didn’t get through nearly as many videos or exercises as I was hoping. At the beginning of the week, I had 6 videos and 5 exercises left in the unit Derivatives: Chain Rule and Other Advanced Topics and only got through 3 videos and 1 exercise. 😢 That being said, I probably spent >7.5 hours working through questions this week so I feel good about my effort. In fact, I usually only do KA in the morning but this week there were a few nights where jumped back on in the evening and worked through questions which never happens!

Before I begin, I should mention I got a TON of practice this week using derivative rules (i.e. the power, product, quotient, and chain rules), algebra and BEDMAS, trigonometry (specifically I got more practice finding the derivatives of Secant, Cosecant, and Cotangent), exponent rules, and logarithms. All of the questions I worked through had multiple steps and used combinations of everything I just mentioned. Each question took me at least 5 minutes or longer to work through and often took up at least 1 if not 2 pages in my notes. Working through these questions gave me a lot more confidence using all of these different types of math. 💪🏼💪🏼

Here are two screen shots of logarithm rules and the derivatives of Secant, Cosecant, and Cotangent which helped me this week:

I began the week working through an exercise called Differentiating using multiple rules which, as the title suggests, gave me questions where I had to find the derivatives of composite functions and apply multiple derivative rules to solve them. It took me awhile to get the hang of how to work through these questions. When I worked through them, I realized I needed to slow down, write neatly, and clearly label each function at the beginning of the question. For example, if a question asked me to find the derivative of F(x) = log(1/1-x), before working through the derivative rules and algebra I would write out:

• F(x) = log(1/1-x) = log((1-x)^-1)
  
  • a(x) = log(x)
  • a’(x) = 1/x
  • b(x) = x^-1
  • b’(x) = -x^-2
  • c(x) = 1 – x
  • c’(x) = 1
    
• F(x) = a(b(c(x)))
  • d/dx [F(x)] = a’(b(c(x)) * b’(c(x)) * c’(x)

As you can see, it takes a bit of algebra simply to identify the individual functions within the composite function. After I’d rewrite derivative using Leibniz’s notation, I would then plug in the values of the individual functions and their individual derivatives where required and work through the question from there. Once I got the hang of this, the questions weren’t too difficult but they each took a good amount of time to get through. Here’s an example of a question and my work solving it:

I’m not going to give a full explanation of how to solve the following question, but here’s a page from my notes where I worked through solving the derivative of the function h(x) = (2x)csc(x)sex(x):

After I got through that exercise, I reviewed how to find the derivative of a base number that’s a constant being raised to a variable (not sure if that’s the correct way to phrase it, but something like 2^x). Here is the process to do so:

The last thing I started to work on this week was finding what are known as second derivatives. I have a pretty weak understanding of what these are and a very weak understanding of how to solve them. As far as I know, they’re operations that find the derivative of a derivative or, in other words, they find the slope of the slope of a point. Here’s the notation:

I really don’t understand the proper way to use the notation when working through these questions, but here’s an example of a question I worked through in my notes where I was asked to find the second derivative of the function 5ln(x⁵):

I think I may have written it out wrong and the process should have been:

• d²/dx² [5ln(x⁵)] = d/dx [d/dx [5ln(x⁵)]
  • = d/dx [25/x]
  • = -25/x²

Obviously I skipped a few steps, but I think that’s closer to the proper way to write out the notation using Lagrange’s notation, i.e. saying that d²/dx² = d/dx [d/dx [ … ] ].

It would be a miracle if I managed to get through this unit by Wednesday this week, but I’m ok if I don’t finish this unit by the end of the month. As I mentioned at the beginning, I feel like I got a lot of good work done this week and I’m much more concerned with that than getting through a unit per month. That being said, I’m now 65% of the way through Derivatives: Chain Rule and Other Advanced Topics(1040/1600 M.P.) with only 3 videos and 3 exercises left to get through so I’m hoping I’ll be able to get through the unit by the end of the week. As much as I’d prefer to learn a lot each week rather than get through KA quickly, the competitive side of me definitely wants to get through units as fast as possible. 🏃🏻‍♂️💨