I finally got through the unit Derivatives: Definitions and Basic Rules this week and managed to get 15% of the way through the following unit, Derivatives: Chain Rule and Other Advanced Topics. It took me 2 days to get through the former unit but scored 25/25 on the unit test on my first attempt! I was pumped to get a perfect score on my first attempt but, as I talk about below, I didn’t feel completely confident in a lot of my answers. Even still, considering I hadn’t ever learned anything about derivatives before this unit, I was very happy with how I did and glad to move on to the next unit. 

There were 2 questions on the test, questions 18 and 19, that I found tougher than the others and easily could have gotten wrong. The first one, question 18, was simple for the most part and just asked me to use the derivative product rule to find the derivative of f’(x) = sin(x)x^-2. Here’s the page from my notes where I worked through the question: 

This was a multiple chose question and, after working through the question, the solution I got to was f’(x) = cos(x)/x² – sin(x)/2x³ which wasn’t one of the choices given. The closest solution I saw to mine was f’(x) = cos(x)/x² – 2sin(x)/x³ which is when I realized that if there’s a negative exponent on the variable in a term that’s in the numerator you can bring the variable down into the denominator with the exponent and turn the exponent positive BUT, if there’s a coefficient in front of the variable you do NOT bring the coefficient with it and leave it in the numerator:

I got the question correct but would have gotten it wrong if I had been asked to input my answer on my own as opposed to selecting from a given set of answers.

The very next question, number 19, was even more difficult and I still don’t have a firm grasp on how it works:

The first thing that comes to my mind when looking at this question is that x and sin(x) on the left side of the limit are (x₂, y₂) and 0 and π on the right side are (x₁, y₁). I knew that the answer had to be either B) or C) since those were the two answers that had f’(π) and π is an x value whereas 0 is a y value which wouldn’t have made sense to put into the function, but I wasn’t sure if the function in question equaled sin(x) or sin(x)/x. To be honest, I’m still not exactly sure why the correct answer is sin(x) but my gut told me that was the correct answer which it clearly was. Hurray. 😒

(I’m doing my final edit of this post and just realized that I think if the limit had sin(x)/x in the y₂ position, then the answer would have be B) but I’m not 100% sure about this.)

As I said at the beginning of this post, I got through the remainder of the questions without as much difficulty although there were a number of questions where I wasn’t 100% sure of answer. Nonetheless, I’m happy I got a perfect score and feel good about my overall understanding of derivatives and moving on. 

I began the next unit Derivatives: Chain Rule and Other Advanced Topics on Thursday morning and got straight into the definition of the ‘Chain Rule’:

To me, this rule is similar to the derivative power, product, and quotient rules that I’ve learned already. The chain rule is essentially just a formula you can use to find the derivative of a composite function. The key when using this rule is to 1) identify that you’re dealing with a composite function, and 2) figure out what would be considered the ‘inner’ and ‘outer’ functions and label them appropriately. For example:

• F(x) = √cos(x) = [cos(x)]^1/2
  • Inner function is g(x) = cos(x)
  • Outer function is h(x) = x^1/2
  • g'(x) = -sin(x)
  • h’(x) = 1/2√x
• F’(x) = h’(g(x)) * g’(x)
  • = h’(cos(x)) * (-sin(x))
  • = 1/2√cos(x) * (-sin(x))
  • = -sin(x)/(2)√cos(x)

The important thing to remember is that when you’re trying to find h’(g(x)) you substitute the entire expression cos(x) in place of x in the function h(x) = 1/2√x (i.e. 1/2√cos(x)). It’s also important to remember NOT to substitute g’(x) (i.e. 1/2√(-sin(x)).

The last few videos I went through this week showed me how to use Leibniz’s notation when using the chain rule:

About 2/3’s of the way down the page you can see Leibniz’s notation for this particular composite function made up of function a(x) and function b(x):

• f'(x) = d(a)/d(b) * d(b)/d(x)
  • “F-prime of x equals the derivative of function a with respect to function b multiplied by the derivative of function b with respect to x.”

I feel like I have a decent understanding of how this type of notation works, but I definitely find Lagrange’s notation (i.e. f’(x) = a’(b(x)) * b’(x)) easier to understand. I got through the first 3 exercises in the unit which all had to do with using both types of notation to solve for the derivative of composite functions which I found fairly straightforward.

Looking ahead and assuming nothing comes up that stumps me for too long, I think I should be able to get through this unit, Derivatives: Chain Rule and Other Advanced Topics (240/1600 M.P.), within 3 weeks. I’ll be glad if that’s the case since it will keep me going at my ‘1 unit per month’ pace. I’m hoping to get through the next 3 sections of the unit this week which combined have 14 videos but only 3 exercises. I’m a bigger fan of working through exercises than I am of watching videos so I feel like this week could be a grind. Hopefully it’s not too bad!