At the beginning of this week, I had some struggles. We did a worksheet on more chain rule problems, and even though I didn't have any problems with the chain rule, for some reason I was having a lot of problems. First of all I kept taking the chain rule over and over. When I was supposed to just multiply by the derivative of the inside, I starting multiplying the answer by the second derivative. It was a mess and I ended up doing significantly more math than was necessary and making my life hard.
Our new lesson this week was implicit differentiation. Implicit differentiation is when the dependent variable is not isolated and the equation is not a function. You have to derive both sides of the equation, and the derivative of the dependent variable, 'y' in most cases, is a function, so the derivative is dy/dx. It's a great time! And that's only 80% sarcasm! Then, we learned about taking higher order derivatives of implicit equations. So we take the second derivative of the equations. It's pretty straight forward. You just take the derivative and then when you end up with dy/dx, you just substitute in what you just found dy/dx to be!!!!!!! Here's a link to a video explaining implicit differentiation: https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-advanced/ab-implicit-diff/v/implicit-differentiation-1
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This week we learned more rules for solving derivatives. The week's lesson was pretty much just focused on learning the chain rule. It is a pretty simple concept, but when you add multiple chain rules inside each other and mix in other rules for derivatives, it can get a little confusing and complicated. You just take the derivative of the outer function with keeping the inner function intact, and then you multiply that to the derivative of the inner function. This week we learned more about derivatives!!!!!!! WOOT WOOT!!!! We learned a bunch of rules for solving more complex derivatives including for quotient and multiplicative derivatives, trigonometric derivatives, anti-derivatives, and higher order derivatives. I feel pretty good about this unit so far. I have already learned the quotient rule and the product rule by heart from the homework. The one thing that I have been having difficulty with isn't even conceptual- I just have a very hard time remembering the +C that you have to include on the end of antiderivatives. In a derivative there is no evidence to tell you if there is a constant in the original function. This is because constants do not affect the slope, they affect the position of the function in respect to the y-axis, and since the derivative is the slope of the tangent at any point, it only takes into account the parts of the function that affect slope. For this reason the original function of a derivative could have any constant in the function. It is unclear which, so when doing antiderivatives you have to include +C. I need to find a way to remember the +C.
Below I have included a method of rearranging a problem so that you can avoid using the quotient rule over and over again. |
Josephine Swaney's AP Calc Blog Archives
January 2018
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