I struggled again this week and didn’t manage to get through the unit test for Applications of Derivatives. I spent the first 2 days working through the final exercise and didn’t start the test until Thursday. I made one careless mistake on the test which resulted in me getting a single question wrong which was annoying. That being said, there were a number of questions that I couldn’t remember how to solve and had to go back and review my notes so, as disappointed as I was to not pass the test, it’s probably for the best that I’ll have to go back and review some of the videos this coming week before I redo the test.

I spent Tuesday and Wednesday finishing up with L’Hôpital’s Rule and learning how to apply it to composite exponential functions (i.e. when a function is being raised to another function – ex. (x – 1)^{(x – 1)}). Working through the exercise for this section helped me get a stronger grasp on the fact that you have to treat the numerator and the denominator as two separate functions and find the derivatives of each in order to find the limit. This exercise also helped me get better at using all the derivative rules such as the power rule, quotient rule, etc. Here are two questions I went through and how to solve them:

• Q. Find the limit as x approaches ∞ of (3 + 3e^x)^3/x:
  • (Note: The reason why we need to use L’H.R. is because if you input ∞ into x you’re left with ∞^∞ which I think just equals ∞. 🤔)
  • y = (3 + 3e^x)^3/x
    • ln(y) = 3/x * ln(3 + 3e^x)
    • ln(y) = (3 * ln(3 + 3e^x))/x
  • (Note: at this point, I would then write out all the individual functions that make up the composite function)
    • a(x) = ln(x)
    • a’(x) = 1/x
    • b(x) = (3 + 3e^x)
    • b’(x) = 3 e^x
    • c(x) = x
    • c’(x) = 1
    • ln(y) = 3 * a(b(x))/c(x) = 3 * d/dx[a(b(x))/c(x)]
  • (Note: I always have to remind myself that the reason why a(b(x))/c(x) is equal to its derivative, d/dx[a(b(x))/c(x)], is because of L’H.R. which states the limit of 2 functions being divided by each other that equal 0 or ∞ is the limit of their derivatives divided by each other.)
    • ln(y) = 3 * (a’(b(x)) * b’(x))/c’(x)
    • ln(y) = 3 * ((1/(3 + 3e^x) * 3e^x)/1
    • ln(y) = 3 * 3e^x/(3 + 3e^x)
    • ln(y) = 9e^x/(3 + 3e^x)
      • @ x->∞
    • ln(y) = 9e^∞/3e^∞
    • ln(y) = 9/3
    • ln(y) = 3
  • (Note: Now that we’ve established a limit does exist when dividing the derivatives of the two functions, you can take both sides of the equation to e^ in order to eliminate the natural log on the left and solve for the limit as x approaches ∞ of y which equals (3 + 3e^x)^3/x.)
    • e^ln(y) = e^3
    • y = e^3
    • (3 + 3e^x)^3/x = e^3
    • Lim x->∞ (3 + 3e^x)^3/x = e^3

(Note: Looking at the first photo, if you input ∞ into the exponent 1/(x² + 2x) you’re left with 1/∞ which, as far as I know, equals 0, however when you look at section 1/5 in the first photo it states that the exponent ends up being ∞ where it says 1^∞. This really confused me so I Googled it and even asked the Subreddit r/Mathhelp and still didn’t get a clear answer. It seems that 1/0 can be considered infinity in some cases but not in other but I’m not sure exactly how it works.)

• Q. Find the limit as x approaches 0 of (2e^x – 1)^{1/(x^2 + 2x)}:
  • y = (2e^x – 1)^{1/(x^2 + 2x)}
    • ln(y) = 1/(x² + 2x) * ln(2e^x – 1)
    • ln(y) = ln(2e^x – 1)/(x² + 2x)
  • Individual Functions:
    • a(x) = ln(x)
    • a’(x) = 1/x
    • b(x) = 2e^x – 1
    • b’(x) = 2e^x
    • c(x) = x²
    • c’(x) = 2x
    • d(x) = 2x
    • d’(x) = 2
  • Use L’H.R. on the limit/functions:
    • ln(y) = a(b(x))/(c(x) + d(x)) = a’(b(x)) * b’(x)/(c’(x) + d’(x))
    • ln(y) = (1/2e^x – 1) * 2e^x/(2x + 2)
    • ln(y) = (2e^x/2e^x – 1)/(2x + 2)
      • (@ x->0)
    • ln(y) = (2e⁰/2e⁰ – 1)/(2(0) + 2)
    • ln(y) = (2*1/(2*1 – 1))/(0 + 2)
    • ln(y) = (2/1)/2
    • ln(y) = 2/2
    • ln(y) = 1
  • Take both sides to e^ in order to eliminate the natural log on the left side of the equation:
    • e^ln(y) = e¹
    • y = e
    • lim x->0 (2e^x – 1)^{1/(x^2 + 2x)} = e

After getting through the final section of L’H.R. and composite exponential functions, I began the unit test on Thursday which took me until Friday to finish and, as I mentioned at the top, I only got 1 question wrong. Although I was struggling to understand how to do the question, I actually would have got it correct but simply forgot to add a negative sign in the final step of the question. I’m not going to go through the math, but here’s the question from my notes:

This question involves “differentiating related functions” which, after going back and trying to redo the exercise, I realized I don’t have a very good understanding of. I was hoping I would be able to quickly redo the exercise and bring my score back up to 80/100 on this section (which is what the score needs to be in order to do the unit test and get 100% on the entire unit) but when I tried doing that exercise I failed miserably, so I’ll have to spend the first few days of this coming week reviewing that entire section. (That was a long sentence.)

Assuming I can get through that section on differentiating related functions in the first few days of next week, I think should be able to finish off this unit by the end of next week and finally move on to Analyzing Functions (0/1,800 M.P.). I feel like derivatives are still over my head and that I don’t really know how they work. I’m reminding myself that I felt the exact same way when I worked through Trigonometry and the unit circle and I eventually got a much better grasp on how those concepts. I’m really hoping that’s what happens and derivatives and Calculus, in general, but I’m a bit worried considering how out of my depth I feel. As always, I’m just going to keep grinding away and am going to assume it’ll all eventually make sense at some point! 😤