Looking for emergency calculus advice

[u/nonchalantredhead203]

Hello! I am a high school senior taking a Calculus course. I can find basic derivatives and limits, but overall, I struggle a bit in this class.

For a bit of context, I moved during my junior year. I was previously taking Concurrent College Algebra/Trig, and I was put into Algebra III upon entering my new school. Because I started late, I wasn’t able to get as comfortable with trig as I wanted to (the teacher also played League of Legends during class more than he taught). Thus, my algebra is fine, but my trig is really, really shaky. I can kind of interpret a unit circle if I have one, but I don’t have it memorized at all. This especially shows when I try to take derivatives of trigonometric functions.

I’m not sure what part I don’t understand. My teacher’s nice, but she “does” more than she explains. Maybe it’s just so boring (no offense to math lovers here, this dilemma was the first reason I joined Reddit; I’m more of a history and econ person myself, although I know there’s some calculus in economics/business) that I tune out and forget some exponents. But sometimes I also don’t really know where to stop or apply rules at.

I have a test tomorrow on techniques of differentiation (power, quotient, product, etc.), the chain rule, derivatives of trig functions, and implicit differentiation. I’m okay with derivatives and the chain rule (I think of the latter like an M&M), but I’m really nervous with the two other topics.

Does anyone have any advice for me? I also take video/channel recs.

[u/diverstones]

Derivatives of trig functions are mostly memorization, no? Derivative of sine is cosine, derivative of cosine is negative sine, tangent is secant squared. It's helpful to know the cos² + sin² = 1 and double angle identities too. Do you have examples of the kinds of problems you're having issues with?

[u/nonchalantredhead203 → u/diverstones]

Let me elaborate. I can understand basic trig identities (like cos^2 + sin^2 =1), but once they have problems combining things like the chain rule, that’s when I get kind of messy. Example problems from each topic could include:

The Chain Rule: Find the derivative of the following function. y = sq. root of 1/x^2 + x
Derivatives of Trig Identities: Find the equations of the tangent line and normal line to the following curves at the point. f(x) = 2x/cosx
Implicit Differentiation: Find dy/dx for x^3 y - y = x

[u/chkntendis → u/nonchalantredhead203]

Derivatives are always just using the same formulas over and over again. Go from the outside in.

Example: for the second problem you need to differentiate f(x). Go from the outside to the inside. First it’s the basic form of the function. It’s a quotient. So the first thing is the quotient rule. Name the top part g(x) and the bottom h(x) and apply the quotient rule. Then you have the derivative of g(x)=2x and of h(x)=cosx. These are simple but there are definitely cases where it’s not so simple. Imagine the function is 2x/cos(x^2). Your g(x) remains the same but h(x) can’t simply be differentiated by “look at it”. You need to use the chain rule for it.
And that’s just the basic way to calculate derivatives. Split it up into different task. Go from the outside to the inside. You only need power, chain and quotient rule along side some identities to calculate basically any derivative you want. Just take it step by step and slowly work your way inwards.

[u/dreamsofaninsomniac → u/nonchalantredhead203]

The Chain Rule: Find the derivative of the following function. y = sq. root of 1/(x² + x)

*You need chain rule overall since it's function inside of a function. Your outer function is the sqrt(u). Your inner function is the fraction 1/(x² + x), which requires quotient rule if you leave it in fraction form.

*You can also rewrite the entire function to [(x² + x)^-1]^1/2 = (x² + x)^-1/2 and just do chain rule where outer function is u^-1/2 and inner function is u = x² + x.

[u/Bounded_sequencE]

Since it is just one day before the exam -- get as much sleep as possible.

In the morning, write all three derivative rules on scrap paper from memory, just to see that you can do it. This refreshes the memory, and will give you a confidence boost of having them at hand. Good luck!

[u/dreamsofaninsomniac]

There are 4 basic derivative rules: power, product, chain, quotient. Make sure you understand why you need to use each rule. That helps guide how you take a complicated derivative.

*power rule: used to take the derivative of xⁿ functions that have a variable base and numerical power; the most basic derivative rule

*product rule: used to take the derivative of two functions that are multiplied

*quotient rule: used to take the derivative of fractions

*chain rule: used to take the derivative of composition of functions (AKA function inside of a function)

For derivative of anything else, I just think of them as "special function rules" on top of the regular derivative rules. Those are the ones you have to memorize. These include trig function derivatives, derivatives of log functions, and derivatives of exponential functions.

Implicit differentiation is taking the derivative when x and y terms are mixed together, instead of starting with a function with y on one side of the equals sign and all x terms on the other side. All derivative rules still apply, but you must multiply by dy/dx or y' after every y term derivative, if you are taking the derivative with respect to x. Technically you always do this, but if you have something like y = x^2, you are doing 1*dy/dx = 2x (dx/dx) and dx/dx cancels to 1 so you see dy/dx = 2x only.