Calculus l - Differentiation

What is a Derivative? 

The derivative is a function that is derived from the original function using the process of differentiation. It is used to find the slope at some x-value. When the derivative is graphed with respect to its x-value, it can be used to analyze the slope of a line tangent to the point that exists on the original function’s graph. Whenever the derivative is negative, we know that the slope of the tangent line during that interval [x1, x2] is also negative (going downwards) and vice versa.  

Differentiation Notations: 

 

Definition of the derivative: 

 

Basic rules: 

Constant multiplied to a function rule: 

 

 

For single variable polynomials:  
 

Adding two functions: 

 

Subtracting two functions: 

 

 

Helpful-to-have-memorized: 

         

Trig derivatives: 

 

 

Formula for derivatives of natural logs: 

 

Formula for derivatives of logarithms: 

 

Formula for derivatives of exponentials: 

 

 

 

 

 
 

***Remember: a function is continuous if it is differentiable, but not necessarily differentiable if it is continuous.*** 

Jump discontinuities are always non-differentiable, but they may have one-sided derivatives. Wherever there is a vertical line, it’ll be non-differentiable. Sharp turns are undifferentiable because the left and right side derivatives don’t match, so we don’t know which way the tangent line would be angled at that point.  

 

 

 
 

 

 
 

Here’s a modified one you can use when you have a function inside a function inside a function... 

 

 

Implicit differentiation: used when x and y are not related in a simple manner like    x = y    or   y = f(x). 

 
 

However, here is an example when you would not really need to use the quotient rule: 
 

 

 

 

 
 
Some tougher examples I did for fun 😊:  

 

                                                              

 

 

Calculus I - Limits and Continuity

I've decided to venture out and take an online course via SAIL. You guessed it, Calculus! I'm excited about this one, so I hope you are too. 

 Limits and Continuity 

Limits:

Do not care about what happens at a point, simply care about what happens as you get infinitely closer to the point.

Methods to find the limit:

1.) Plug in the value the limit approaches. 

2.) If an error occurs, try:

- Factoring

- Mulitiplying by the conjugate

- Foiling or Expanding

- Creating common denominators

- For indeterminant forms (0/0 or inf/inf) use L'Hopital's Rule [below]

L'Hopitals Rule: derive the numerator and denominator (keep it in fractional form), then plug in the value the limit approaches. If after plugging in the value, you come back to the original limit, that means that there may be no proper conclusion. 

Properties of Limits:

Rules for limit as x approaches 0 for sin(x) / x:

Limit as x approaches 0 for (sin(x)/ x) is always 1.

Limit as x approaches 0 for (sin(ax)/ sin(bx)) is always a/b. 

Limit as x approaches 0 for (sin(ax)/bx) is always a/b. 

Limit as x approaches 0 for sin(1/x) is always non-existent. We don't really know what happens at x = 0.

Limits that never exist:

Limit as x approaches 0 (1/x^2). However, some may argue this is infinity. 

Limit as x approaches 0 for sin(1/x)

Limit as x approaches 0 for x/|x| (left and right limits will never 

Continuity

Types:

Removable Discontinuity:

If you have get a zero in the denominator (undefined) and you can use a method to get rid of the "undefined" , then you have a removable discontinuity. This would appear as an open circle on the graph:

Jump Discontinuity:

The left and right side limits exist, and are finite, but aren't equal. You will find the graph approaches the same x-value at two different y-values:

Infinite Discontinuity:

Either one or both the left and right side limits do no exist or are infinite. You will find that one or both sides will go to plus or minus infinity, or it will be terribly hard to figure out what is happening at a point:

Rules for Continuity:

For a function to be continuous at "c":

- f(c) should be defined at c. 

- Limit as x approaches c of f(x) should exist.

- f(c) = limit as x approaches 0 of f(x). 

Intermediate Value Theorem:

At an interval [x1, x2], if the function is continous, the function should be defined (have y-values for all x-values) within that interval. 

Factorials - Permutations and Combinations

Combination: probabilities selections where the order DOES NOT MATTER.

Permutation: probabilities where the order MATTERS.

Formulas:

Why is 0! = 1?

Say we take permutation formula from above, and set n = 3 and r = 3. The permutation formula with these number inputs would be:

P(3,3) = (3!)/(3-3)!

This equals 3! which means the denominator must've been 1. 

Another way to approach it:

_ _ _

We have above 3 possible slots, and we only have 3 options (A, B, C). So, our possible outcomes (when order matters) would be as follows:

A B C

B C A

C A B

B A C

C B A

A C B

*if we put the first option at the back and scoot everything over by 1 again we'll just have made a repetitive set*

B A C

Anyways, we now see we have 6 options which is indeed 3! = 1 * 2 * 3

Therefore, 0! = 1.

 :)

Math/CS blog made from an intrigued 11th grader perspective.

(3-5 minute read)

Hello world! Glad you found my blog. Not quite sure about the journey you took to get here (as there's only a 1 in about 2 billion probability of landing on this page) unless you intended to search it up because I told you!! Nevertheless, hope you choose to stay. I'm here to try to make my love for math (even on the worst days) contagious. If you find yourself crying over a problem, just remember it means you actually care and that's true interest [dedication]. So, I completely get it if math isn't your favored or strongest subject; I designed this website to provide several ways of memorizing/solving math problems, with some examples from real text-books/ websites. If anything here helps you out, or you would like to ask a question I am more than happy to respond back to your comments. 

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