This week we learned about the chain rule, u substitution, implicit differentiation and higher order implicit differentiation. The chain says that when you take the derivative of a composite you first take the derivative of the outside and then multiply that by the derivative of the inside. Mathematically that is (fog)'(x)=f'(g(x))•g'(x). It looks more complicated than it really is. U substitution is for finding antiderivatives of composites. Basically you plug in u for whatever is in the parenthesis then find the derivative and find the derivative of u. I understand how it works it's just hard to explain in words. Once I got the hang of these u substitution problems they were kind of fun to do in the homework. Implicit differentiation is when you have y in the problem such as x+2y=5x-y. For this there is like four steps you have to do. First you have to take the derivative of both sides while treating y as a function. Then collect all the terms with y', which is equal to dy/dx, onto one side of the function. After that you have to factor out the y' and then do the algebra so the function is set equal to y'. For the higher order implicit differentiation problems all you have to do is do implicit differentiation as many times as it asks you to. These concepts didn't seem very hard to me but if you miss one step or make one little mistake then it screws up the entire problem. The quiz was somewhat challenging. Number nine was really a struggle. Overall I think I have a pretty good understanding of the concepts though. Hopefully the test won't be too challenging. There was a problem on the quiz with a circle that was kind of confusing but when the equation was written on the board it made it a lot easier to solve.

This week we learned about higher order derivatives, the derivatives of trig functions, and the chain rule. The higher order derivatives we talked about had to do with position, velocity, acceleration, and jerk. Basically you start with position and then take the derivative as many times as you need, like once for velocity, twice for acceleration, and three times for jerk. Next we learned about the derivatives of trig functions. These are pretty easy but it's just hard to memorize all of them so I wrote them on my cheat sheet. I will most likely have them memorized soon if we keep working with them, which I'm assuming we will.We took a quiz on 3.3 and 3.5 and I think that I did pretty good. None of the questions seemed really hard to me so hopefully I got a good score. I think that I did. After the quiz we started to learn about the chain rule. Basically the chain rule is

(f o g)'(x)=f'(g(x))•g'(x). In simpler terms you just have to take the derivative of the outside and just ignore the inside, then multiply that by the derivative of the inside. This concept was fairly easy and I understood the homework pretty well. Then on Friday we learned about how to use the chain rule and u substitution to find the antiderivatives of composite functions. This was confusing. I watched the video and then tried to do the worksheet but I struggled. I understand what u is but and I get how to take the derivative of u but then substituting u in is where I get lost. I don't really understand how that works. I am going to try and do the worksheet and see if it sarts to make some more sense. I will probably try and watch the video over again too. That should hopefully help me out a bit.

This week we continued to learn about derivatives and how they work. We learned about graphing the derivative and what the graph means. When the graph of the derivative is below the x-axis then that means that the original graph is decreasing. When the derivative is above the x-axis the original function is increasing. The zeros of the derivative graph are the maxima and minima of the original graph. This I a concept that I grasped pretty well. After looking at multiple examples it really made sense to me. We also learned about other ways to denote a derivative. The most common is dy/dx which reads "the derivative of y with respect to x." Also there is d/dx which reads "the derivative of f at x." Next we learned about where a derivative fails to exist. Those places are corners, cusps, vertical tangent, and discontinuities. This was a good thing to know because on the quiz there was a question that dealt with several of these things. Next we learned a shortcut to finding the derivative which would have been helpful when we first started learning about derivatives. The short cut is this... f(x)=x^n then f'(x)=xn^(n-1). That makes it really simple to quickly find the derivative of any function. Also we learned about the nderive feature on the calculator. This makes it easy to graph a derivative or find the derivative at a point. Then we took a quiz which I got a 28.5/30 so I was pretty happy. Today we learned about the product and quotient rules for derivatives. So far those don't really make much sense but I haven't had any practice with them yet so I'm sure they will make sense soon. The product rule reads... d/dx(u•v)=dv/dx(u)+du/dx(v). The quotient rule reads... d/dx (u/v)=( du/dx(v)-dv/dx(u))/v^2. That one is pretty complicated and doesn't make much sense yet. Finally we learned about higher order derivatives which is basically just taking the derivative of a derivative. It's pretty easy. Overall this was a pretty productive week I would say.

This week we dove into chapter 3 and derivatives. First we learned what a derivative was. It's the slope of any given on a function. We learned about taking the slope of tangent lines. We learned that to find the derivative at a point you have to use the limit as h approaches 0 of f(a+h)-f(a)/h, where a is any given x value. To find the derivative of a function you have to use the limit as h approaches 0 of f(x+h)-f(x)/h. We did a lot or practice with this and we did several worksheets to go with it too. There was a four step process to go along with this. First find f(x+h). Then write f(x+h)-f(x) and simplify. Next find f(x+h)-f(x)/h and simplify. Finally find the limit as h approaches 0 of f(x+h)-f(x)/h. That will give you the derivative of the function. This week I understood pretty much everything. At first things were a little sketchy but once I figured it all out it got a lot easier. One thing I struggled with a little bit was understanding the graphs of the derivatives. It was hard to understand what they meant but after awhile I started to figure it out. Also we had to cube x+h quite a few times, and luckily Ishan reminded me and Kiegan of Pascal's triangle and that made it much easier to do the cubing. I think that this week was pretty important because we will probably have to use derivatives in the future. It's kind of a calculus thing. And limits came up again this chapter so I'm starting to see how all the things we learn will kind of connect. Math class is kind of like a toolbox that we keep adding tools to every week.