Calculus 1211 Learning Journal Unit 4 PDF

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Summary

Consider one mathematical idea from the topics this week that you have found beautiful and explain why it is beautiful to you. Your answer should:
explain the idea in a way that could be understood by a classmate who has taken classes X and Y but has not yet taken this class
address ho...

Description

This week we have studied many beautiful topics. One the topics I have found beautiful, it was the derivative of quotient and its rule. First, let’s know a little explanation about derivative and how you should calculate: In order to calculate derivatives, we have two options: depending on whether one uses the definition of a derivative, which is a limit that tends to absolute zero, or uses rules of derivation, whose operation is assured. As a first step, when derivatives exist, they calculate the slope of the tangent line to a function f(x). All kinds of mathematical problems can be solved by determining the slope, also called rate of change. The most used notations for the derivative of the function f (x) are: f ' (x) or [f (x)]'. If these derivatives are calculated at point p, the notations will be: f '(p) or [f (p)]'. Calculating this limit is not a big challenge for polynomial functions with degree 2 or 3, since the properties of limits ensure that the limit of the sums is equal to the sum of the limits and, therefore, given the limit of a polynomial, it is enough to calculate the limits of each monomium that formed it. However, very high degree polynomial functions or other types of functions impose a high degree of difficulty in calculating this limit. Thus, seeking greater flexibility and ease in calculating derivatives, it is possible to prove the subsequent results, usually known as properties of derivatives, or rules of derivation. Below is a list of all the derivative rules we went over in class. • Power Rule: f(x) = xn then f (x) = nxn−1 • Product Rule: h(x) = f(x)g(x) then h(x) = f(x)g(x) + f(x) g(x) • Quotient Rule: h(x) = f(x)g(x) then h (x) = f 0(x)g(x) − f(x) g(x) g(x)2 • Chain Rule: h(x) = f(g(x)) then h 0(x) = f0 (g(x))g0(x)

I found more beautiful the topic about the Quotient Rule. In calculus, the quotient rule is used to find the derivative of a function that has a ratio between two differentiable functions. Example of Quotient Rule

( f/g )’ = gf’ − fg’ /g2

The derivative of "High over Low" is: "Low dHigh minus High dLow, over the line and square the Low" Example: What is the derivative of cos(x)/x ? In our case: f = cos g = x We know (from the table above):

f' = −sin(x) g' = 1 So: the derivative of cos(x) x= Low dHigh minus High dLow square the Low= x(−sin(x)) − cos(x)(1) x2= − xsin(x) + cos(x)x2

Reference: List of derivatives rules (2020),https://www.math.ucdavis.edu/~kouba/Math17BHWDIRECTORY/Derivatives.pdf Herman, E. & Strang, G. (2020). Calculus volume 1. OpenStacks. Rice University

Kuta, Software LLC (2017) . Retrieved from:http://bhsapcalculus.weebly.com/uploads/7/7/1/1/7711905/2_derivatives__sum_power_product_quotient_chain_rules.pdf Derivate,rules. (2021) https://www.mathsisfun.com/calculus/derivatives-rules.html...