It was a disappointing week. Not only did I not get through the unit Derivatives: Definitions and Basic Rules, but I didn’t even start the unit test. I did manage to get everything finished besides the test, however, and actually got much better at understanding trig functions, but, regardless, I’m still very disappointed that I didn’t get through the unit test. The reason I didn’t get as far as I wanted to is because I was having some random health issues that made it incredibly difficult for me to concentrate. I don’t need to get into the details, but the silver lining is I seem to be getting better. I’m getting very tired and frustrated of feeling sick and worried about covid and really just want things to get better.

AASAGGHSDHHUVB!!!!!!! 😡😡😡😡

I picked up this week learning more about the derivative quotient rule. As a recap, the formula for the quotient rule is:

• f(x) = a(x)/b(x)
  • “The function f(x) equals the function a(x) divided by the function b(x).”
• f’(x) = (a’(x)*b(x) – a(x)*b’(x))/[b(x)]^2
  • “The derivative of the function f(x) equals a’(x) times b(x) minus a(x) times b’(x), all divided by b(x)^2.”

I spent all of Tuesday and Wednesday working through the remaining videos and exercises in this section. It was helpful working through these questions to drill in the quotient rule formula but also to get practice using the power rule to find the derivatives of individual terms in the equations. These questions had me use a lot of factoring and exponent properties which was useful practice.

The last remaining section of the unit I had to do with finding the derivatives of Tangent, Cotan, Secant, and Cosecant. As disappointed as I was with the week overall, I was very happy with what I took away from this section of the unit. I finally started to understand Cotangent, Secant, and Cosecant which has always been confusing to me. The fundamental concept that helped me understand Cot, Sec, and Csc is that they’re the reciprocals of Tangent, Cosine, and Sine, respectively:

• Sin(θ) = Opp/Hyp
  • Csc(θ) = 1/(Opp/Hyp)
    • = 1 * (Hyp/Opp)
      • (Note: In order to understand this step, you must remember that dividing any two values is the same thing as multiplying the numerator by the denominator’s reciprocal. Ex. 2 ÷ (4/3) is the same thing as 2 * (3/4).) 
    • = Hyp/Opp
      
• Cos(θ) = Adj/Hyp
  • Sec(θ) = 1/(Adj/Hyp)
    • = 1 * (Hyp/Adj)
    • = Hyp/Adj
      
• Tan(θ) = Opp/Adj
  • Cot(θ) = 1/(Adj/Opp)
    • = 1 * (Opp/Adj)
    • = Opp/Adj

Having a better understanding of how to derive Cosecant, Secant, and Cotangent from Sine, Cosine, and Tanget made what I learned next, finding the derivatives of Tan, Cot, Csc, and Sec, much easier to understand. In order to figure out the derivatives of each, you have to understand a few things; 

1. d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = -sin(x)
2. The derivative power rule
3. The derivative quotient rule
4. a² + b² = c² which is the same thing as cos²(x) + sin²(x) = 1² = 1
5. Secant, Cosecant, and Cotangent are the reciprocals of Sine, Cosine, and Tangent

The key takeaways from the above two photos are:

• d/dx[tan(x)] = 1/cos²(x)
• d/dx[cot(x)] = -1/sin²(x)
• d/dx[sec(x)] = sin(x)/cos²(x)
• d/dx[csc(x)] = -cos(x)/sin²(x)

The rest of my week was spent working through questions that asked me to find the derivatives using the above formulas. Often the questions would say something like, “find the derivative of cot(x) @ x = π/6”:

• d/dx[cot(x)] = d/dx[1/tan(x)]
  •  = d/dx[1/(Opp/Adj)]
  • = d/dx[Adj/Opp]
  •  = d/dx[cos(x)/sin(x)]
  • = ((-sin(x))(sin(x)) – (cos(x))(cos(x)))/sin²(x)
  • = (-sin²(x) –  cos²(x))/sin²(x)
  • = (-1)(sin²(x) + cos²(x))/sin²(x)
  • = (-1)(1)/sin²(x)
  • = -1/sin²(x)
• d/dx[cot(π/6)] = -1/sin²(π/6)
  • = -1/(1/2)²
  • = -1/(1/4)
  • = -1 * (4/1)
  • = -4

There was only one exercise in this section but the exercise had 7 questions and each question required me to go through all of the steps in the example I gave just above. I kept making simple mistakes and it took me 4 or 5 attempts to get through the exercise but, as frustrating as it was, it was really good practice for me to understand the unit circle, 30-60-90 and 45-45-90 triangles, and cementing the derivative power and quotient rules into my head. 

I’m 80% of the way through Derivatives: Definitions and Basic Rules (2000/2500 M.P.) and only have the unit test left to go. I opened it up and saw that it’s 25 questions long (I’m just realizing that each unit test must have 1 question per 100 M.P.) so it might take me a few attempts to pass it with a decent score.  Considering how long it is, if I get 23 or 24 out of 25 correct, I may just move ahead. I’m hoping I can get through the test early next week so I can get started on the following unit Derivatives: Chain Rule and Other Advanced Topics, but what I’d like even more is to FINALLY START FEELING HEALTHY!!!  😭 😭 😭