Even though I spent most of this week working through equations, which is what I typically like doing most on KA, this week was very tough and frustrating for me. I went through 28 pages of notes yet I still feel like I didn’t get through very much. The majority of my week was spent learning about implicit differentiation questions which I’m still having a hard time understanding. By the end of the week, I finally got the hang of how to solve these types of questions but don’t have an intuitive grasp on what’s going on. As frustrating as it is, it’s the type of thing where I assume a lightbulb will go off if I keep working through the equations and exercises. (At least I hope that’s what happens! 😰)
An important thing I realized this week was what the definition of “differentiation” is:
I’m pretty sure this wasn’t explicitly mentioned in the videos I’ve watched so far (which I find somewhat annoying if that’s true) but, as you can see from the above quote, differentiation simply means finding the derivative of an equation.
Up to this point I’ve only worked through what’s known as explicit differentiation which, as far as I understand, is finding the derivative of a function when the function is setup so that the variable y (i.e. f(x)) appears alone on one side of the equation. The problem with explicit differentiation is that it won’t always work to find the derivative of every function. For example, if the function contains a y3, using explicit differentiation won’t work (or will be incredibly difficult, apparently).
The way I understand it, when using implicit differentiation, your aim is to find dy/dx, a.k.a. “the derivative of y with respect to x.” You begin the process but finding the derivative of x, a.k.a. d/dx, of both expressions on either side of the equation. (Reading that back, that may have been hard to understand.) Here’s an example:
- x2 + y2 = 1
- d/dx[x2 + y2] = d/dx[1]
- d/dx[x2] + d/dx[y2] = 0
- 2x + 2y*dy/dx = 0
- (Note: This is the part I still don’t understand. The explanation that KA gives is that “the derivative of y2 is 2y*dy/dx and not simply 2y … because we treat y as a function of x” but I still don’t really understand why you’re supposed to treat y as a function of x.)
- 2y*dy/dx = -2x
- dy/dx = -2x/2y
- dy/dx = -x/y
I watched a few videos from other YouTubers about how/why implicit differentiation works (one, two, three) which helped me understand that if you’re using implicit differentiation, whenever you’re finding the derivative of a term that contains y, you always are left with a dy/dx. Even though I don’t fully understand why it works this way, I feel like I have a good grasp on the process of using implicit differentiation in this regard so I’m hoping that I’ll eventually figure it out the more I practice.
As I mentioned, learning about implicit differentiation took up the majority of my week but I did briefly work through finding the derivatives of inverse functions:
Using the formula f’(x) = 1/g’(f(x)) wasn’t difficult and relatively easy to understand. One thing I didn’t mention in the photos above that helps me think about how inverse functions work is that they simply swap the (x, y) coordinates of the OG function:
- Coordinates of OG function –> Coordinates of Inverse function
- (x, y) –> (y, x)
- (1, 3) –> (3, 1)
- (14, -76) –> (-76, 14)
- (π, 5) –> (5, π)
Even though I didn’t learn too much this week, I feel like implicit differentiation is a crucial component of calc so I’m happy that by the end of the week I started to get the hang of it. I’m now 35% of the way through Derivatives: Chain Rule and Other Advanced Topics (560/1600 M.P.) so it’s getting less likely that I’ll be able to finish this unit off before the end of the month, BUT I do still think it’s possible. I have 17 videos and 9 exercises left to get through plus the unit test. I’m hoping to get through at least 10 videos and 5 exercises this week which would set me up well to finish everything by the end of the month. Hopefully I won’t come up against anything as tricky as implicit differentiation in the next two weeks!