This week I went through 59 pages of notes. 59 pages!!! 🤯 It took me two attempts but I scored 15/15 on the Derivatives: Chain Rule and Other Advanced Topics unit test on Sunday after getting 13/15 on my first attempt on Friday. I would say my grasp on the questions for the most part was pretty good but I need more practice to completely and intuitively understand a few of the concepts. There were also a few questions that I had to double check my notes for before answering so, although I scored 100%, there should be a bit of an asterisk on my score. In any case, I feel good about moving on to the next unit and think that doing so will probably help cement and strengthen what I’ve learned so far.

Before I started the unit test, I still had a few videos and exercises left to get through. I began the week working on finding second derivatives which helped give me better understanding of how the notation d²y/dx² is used. Here’s a question I worked through with my notes just below:

As you can see from my notes (assuming you can read my chicken scratch handwriting), to find a second derivative you begin by finding the derivative of the function (i.e. d/dx [y⁴ + 5x] = d/dx [11]) and then, once you’ve solved for the first derivative, repeat the same process on the first derivative (i.e. d/dx [dy/dx] = d/dx [-5/4y³]). When finding d/dx [dy/dx], I like to think of it as multiplying the terms in both numerators together and the terms in both denominators together:

• d/dx [dy/dx] = d­­₁/(dx)₁ * d₂y/(dx)₂
  • = d₁  * d₂y / ((dx)₁ * (dx)₂)
  • = d²y/(dx)²

I really don’t know if that’s exactly what’s happening with the notation of d²y/dx² but that’s what seems to make sense to me at this point.

After finishing the exercise on second derivatives, I learned how to find the derivative of composite exponential functions, i.e. when a variable is raised to the power of a variable (ex. y = x^x). The key thing to know when differentiating composite exponential functions is that you need to need to find the natural log of both sides of the equation so that you can bring the power down as a coefficient. This is based off the log rule log_a(b^c) = c * log_a(b). Here is a question I worked through on KA with my notes below:

As I mentioned in my intro, I first attempted the unit test on Friday but got 2 questions wrong. The second question of the test had to do with composite inverse functions and I managed to get it correct but had to leave the test and review a few videos before answering the question. I forgot that in a composite inverse function (ex. f(g(x)) where g(x) is the inverse of f(x)), when you input x you output x, as well (ex. f(g(2)) = 2). Knowing this, you then use the chain rule in order to find the derivatives of both functions. I’m not going to do the algebra but the resulting equation is g’(x) = 1/f’(g(x)).

The first question I got wrong was question 6 which asked me to find the derivative of arctan(-x/2) where x = (-7). First off, I had to look up the equation for the derivative of arctan(x) so right off the bat there was going to be an asterisk beside my answer. Even after looking up the formula, I then I forgot that I needed to apply the chain rule so I ended up getting it wrong anyways. Here’s what I did versus what I should have done:

• What I did:
  • h(x) = arctan(-x/2)
  • h’(x) = 1/(1 + x²)
    • h’(-7/2) = 1/(1 + (-7/2)²)
      • = 1/(1 + 49/4)
      • = 1/(53/4)
      • = 1 * 4/53
      • = 4/53
        
• What I should have done:
  • h(x) = arctan(-x/2)
    • = (a(b(x))
      • a(x) = arctan(x)
      • a’(x) = 1/(1 + x²)
      • b(x) = (-x/2)
      • b’(x) = (-1/2)
  • h’(x) = a’(b(x)) * b’(x)
    • = 1/(1 + (-x/2)²) * (-1/2)
  • h’(-7) = 1/(1 + (-(-7)/2)²) * (-1/2)
    • = -1/(2 * (1 + 49/4))
    • = -1/(2 * (53/4))
    • = -1/(53/2)
    • = -1 * 2/53
    • = -2/53

Incase none of what I just wrote made sense, here’s how it was explained on KA:

After I got that question wrong, I still made an effort throughout the rest of the test but, since I knew I was going to have to do it again, I lost a bit of my focus. I ended up getting question 11 wrong by making a simple mistake of forgetting to multiply one of the terms in an equation by an x:

After getting those 2 questions wrong, I had to go back and redo 2 exercises in order to bring both their scores back up to 80/100 M.P. which I needed to prior to redoing the unit test otherwise I couldn’t have scored 100% on the entire unit.  Both those exercises took me all of Saturday to get through which was annoying but also helpful to review.

It took me ~1.5 hours to finish the test on Sunday and I had to double check a few questions. Question 10 was another composite inverse function which I double checked that I had used the proper formula before answering but, as it turned out, had solved the question correctly before checking. Question 11 was a composite exponential function that asked me to find the derivative of y = x^tan(x) which I found confusing. After Googling it and reviewing my notes, I was able to figure it out:

Lastly, I also would have gotten question 14 wrong if I hadn’t looked it up. The question asked me to find the derivative of h(x) = -3tan³(sin(x)). I thought it was a composite function where the -3 was a coefficient in front of tan³sin(x) which meant I could remove the -3 when finding the derivative (i.e I thought the function looked like h(x) = -3(tan(sin(x))³) and therefore h’(x) = -3 * d/dx[tan(sin(x))³]). It turns out that the outer function should be read as -3x³, as in -3(tan(sin(x))³, and therefore the power rule needs to be applied resulting in the outer function equalling -9x² or -9(tan(sin(x))². 

(I just read that over and it is mad confusing but I’m pretty sure it all makes sense. 😵‍💫)

This coming week I’ll be starting the unit Applications of Derivatives (0/1500 M.P.) which sounds really interesting since, even though I’ve been working on derivatives for a few weeks and understand that they calculate the slope of a point on a function, I don’t really know what practical uses they have (but I’m sure there are many!). Being that the unit is 1500 M.P. and I’m already 5 days into July, I think it’s unlikely that I’ll get through the unit before the end of the month. Considering I spent an extra ~1.5 hours on Sunday working through the unit test and still managed to write my blog post, I’m thinking I may start working through KA on Sundays which would make it more likely that I could finish this unit before August. As I’ve said a bajillion times however, as long as I’m putting in a decent effort each week, I don’t really care how long it takes me to get through each unit. Slow and steady baby, slow and steady! 🐢💨