New Images for Differentiation

Differentiation.md

@@ -1,4 +1,17 @@
-# Overview of what differentiation is:
+[<- Back to index page](https://cpawley.github.io/HHG2-MSP-Physics/)
+
+# Differentiation 
+
+[ ## Overview-of-Differentiation](## Overview-of-Differentiation)
+#### Different notations 
+#### Important Differentiation Rule 
+#### Rules for Combined Functions 
+#### Applications of Derivatives to Real Life 
+#### Glossary 
+#### Exercises 
+#### Additional Video Links
+
+## Overview of Differentiation
 
 Differentiation is, by definition, the way to find a function that represents the rate of change of one variable with respect to another variable. To begin simply, we will take a variable that changes across time, like a car traveling along a road for a defined amount of time. By differentiating this, we can find the velocity. By further differentiating that, the acceleration can be found. Why this is will be further discussed below:
 
@@ -8,13 +21,216 @@ The next step involves understanding what the slope is, and how it can be rewrit
 
 It won't always be as easy as this though, as most graphs in physics are curved, which makes getting the slope much harder to get. In this way, a tangential line is used to represent the slope at a given point. The definition of a tangent is something that only just touches the line, such as in the image below. The point that it intersects at is infinitesimally small, and will from now on be referred to as “d” instead of delta (which represents a finite difference). Therefor, the slope at a tangential point on the graph of position versus time will be written as dxdt; a differentiation. 
 
-Some good websites to read about the laws of differentiation, and practice are as follows:
-Differentiation and Integration - Academic Skills Kit
- (ASK)https://brilliant.org/wiki/derivative-by-first-principle/
-https://en.wikibooks.org/wiki/Calculus/Differentiation/Differentiation_Defined#What_is_Differentiation?
+## Different Notations:
+
+In many study materials, you will encounter variations of the notations of **derivatives**. These different notations are denoted below, so you can recognise them.
+
+* **Leibniz’s notation:** dydxordfdx or ddxf for the first derivative of a function y or f dependent    on x. The second derivatives are d2ydx2,d2fdx2, d2dx2f and  respectively.
+
+* **Lagrange’s notation:** f’ for the first derivative and f’’  for the second derivative (any    
+derivative beyond the third derivative - f’’’ - is denoted as fnfor the nth derivative).
+
+* **Newton’s notation:** ẏ for the first derivative and ÿ for the second.
+
+* **Euler’s notation:** Dxyor Dxf(x)for the first derivative; Dxnf(x)  for the nth derivative.
+
+## Important Differentiation Rule:
+
+When a function is derived with respect to a variable, this general rule applies for finding the derivative:
+**For *f(x) = axn→ f’(x) = naxn-1* **
+
+## Rules for Combined Functions:
+
+* **Constant rule**: If a function f(x) is constant, then its derivative is zero.
+
+*f(x) = c where c is a constant, then: f’(x) = 0.*
+
+* **Sum rule:** When a function f is the summation of two functions g and h, then the derivative of f is equal to the sum of the derivatives of g and h.
+*f = αg + βh →  f’ = (αg + βh)’ = αg’ + βh’.*
+
+* **Product rule:** When a function f is the product of two functions g and h, the derivative of f is the summation of the product of g and h’ and the product of g’ and h.
+*f = gh →  f’ = gh’ + g’h.*
+
+* **Quotient rule:** This slightly more difficult rule is used to find the derivative of a function f that is the quotient of two functions g and h.
+*f = gh→  f’ = g'h-gh'h2*
+
+* **Chain rule:** This rule is used for composite functions, like, for example f(x) = h(g(x)). So, h is a function of g and g in turn is a function of x. The derivative of such a composite function is:
+*f’(x) = h’(g(x))g’(x).*
+
+## Applications of derivatives in real life
+
+*Business:* in business, differentiation is used to determine the profit and loss using graphs.
+
+*Optimisation:* in optimisation, differentiation is used on many occasions. Say, for example, you have a limited amount of material to produce a cylinder that holds liquid. Differentiation can help you to find out how to use the material to make a cylinder that is capable of holding the maximum amount of liquid.
+
+*Physics:* by using differentiation, a moving body’s position allows you to calculate its velocity and acceleration.
+
+*Chemistry:* in chemistry, the change in concentration of an element during a chemical reaction can be estimated using differentiation. Additionally, the rate of the reaction is found using derivatives.
+
+
+Below is a list of common derivatives.
+
+<img align= "center"  src="https://github.com/cpawley/HHG2-MSP-Physics/blob/master/SadVries_images/Derrivatives 1.png?raw=true">
+
+Sometimes, it can be hard to imagine how a certain graph looks yet having an image in mind may prove to be useful when working with derivatives. Below are a few graphs corresponding to some standard derivatives.
+
+<img align= "left"  src="https://github.com/cpawley/HHG2-MSP-Physics/blob/master/SadVries_images/image 2.png?raw=true">
+
+<img align= "right"  src="https://github.com/cpawley/HHG2-MSP-Physics/blob/master/SadVries_images/image 3.png?raw=true">
+
+**Graph 1**: y = -sin(x) which is the derivative of the cosine function. **Graph 2**: y = 1/x which is the derivative of the natural logarithm (ln).
+
+<img align= "right"  src="https://github.com/cpawley/HHG2-MSP-Physics/blob/master/SadVries_images/image 4.png?raw=true">
+
+**Graph 3**: y is a cubic function, which is the derivative of a fourth-degree polynomial.
+
+<span style="background-color:yellow">
+
+Some Markdown text with <span style="color:blue">some *blue* text</span>.
+
+>span style="color:blue"> ##**Sample exercises** >/span>
+
+
+Compute the following derivatives:
+
+1.     f(x) = 4x4 + 3x2 + 1
+
+2.     f(x) = x sin(2x)
+
+3.     f(x) = exx2
+
+4.     f(x) = ln(x2)
+
+</span>
+
+<span style="background-color:light-yellow">
+
+*Answers:*
+
+1. f’(x) = 16x3 + 6x
+
+2. f’(x) = sin(2x) + 2xcos(2x)    (Product + Chain Rules) 
+
+3. f’(x) = x2ex-2xexx4 = xex(x-2)x4= ex(x-2)x3    (Quotient rule)                                           
+
+4. f’(x) = 1x22x =2x   
+
+</span>
+
+## Partial derivatives:
+
+Not all functions consist of only one variable; in many cases, it has multiple. This is were **partial differentiation** comes into play. Let’s use some function f(x,y,z) = 3x2 + sin(xy) + 2z which is a function of three variables: x, y, and z. **When taking a partial derivative, you differentiate the function with respect to only one of these variables; the other variables will become constants.** So, if one were to take the derivative of f(x,y,z) with respect to z, 3x2 and sin(xy) would be treated as constants. As mentioned before, the derivative of a constant is zero. Thus, the derivative will simply be 2. If one would take the derivative with respect to x, the y in sin(xy) is a constant, and the situation is similar to if you would be taking the derivative of, for example, sin(4x). The derivative of f with respect to x is then 6x + ycos(xy). Similarly, the derivative of f with respect to y is xcos(xy).
+
+The partial derivative with respect to x is denoted as f(x,y,z). So, the lowercase delta is used to denote a partial derivative.
+
+You can also take a second partial derivative, which is a partial derivative of a partial derivative. The first partial derivatives with respect to x, y, and z are respectively:
+
+> f(x,y,z) = 6x + ycos(xy)
+
+> f(x,y,z) = xcos(xy)
+
+> f(x,y,z) = 2 .
+
+No matter with respect to what variable you take a second partial derivative of z f(x,y,z), for this example function, it will always be zero. Differentiating xf(x,y,z) and yf(x,y,z) with respect to z will also result in zero, because there are only constants in these partial derivatives. xf(x,y,z) and yf(x,y,z), however, can both be differentiated with respect to x as well as y.
+
+If the former is differentiated with respect to x, that is denoted as:
+
+> 2x2f(x,y,z) .
+
+If you take the derivative with respect to y, it is denoted as:
+
+> 2ydxf(x,y,z) .
+
+> 2ydxf(x,y,z) = 6 – y2sin(xy)
+
+For 2ydxf(x,y,z), the product rule is needed.
+
+> 2ydxf(x,y,z) = cos(xy) – xysin(xy) .
+
+If you would differentiate yf(x,y,z) with respect to x, the result is most likely equal to that of
+2ydxf(x,y,z). 
+
+In most cases, the order in which the two derivatives are taken have no influence on the result. However, since there are functions for which this is not the case, you should not just assume that
+
+> 2ydxf(x,y,z) = 2xdyf(x,y,z) .
+
+## Small increment approximation
+
+A useful application of (partial) differentiation is **small increment approximation.** Small changes in some function f of x and y can be approximated using the following formula:
+
+**δf = (∂ f)/(∂ x)δx + (∂ f)/(∂ y)δy**
+
+Let’s look at small increment approximation using the example of the volume of a cylinder, which is  πr2h. So, the volume is dependent on two variables: the radius r and the height h. When both variables change by a very small amount, you can approximate the change in the cylinder’s volume. 
+
+> δV = (∂ V)/(∂ r)δr + (∂ V)/(∂ h)δh
+
+(∂ V)/(∂ r) and (∂ V)/(∂ h) are the rates of change of V with respect to r and h, respectively; δr and δh are the changes in r and h, respectively.
+
+#### Sample Exercise:
+
+You are baking a cake that is rapidly rising in the oven. Initially, the cake had a radius of 15 cm and a height of 10 cm. At a certain point, the cake has a radius of 15.4 cm and a height of 10.12 cm. Find the approximate change in the cake’s volume.
+
+The equations:
+
+> δV = (∂ V)/(∂ r)δr + (∂ V)/(∂ h)δh
+
+> V = πr2h
+
+> (∂ V)/(∂ r) = 2πrh
+
+> (∂ V)/(∂ h) = πr2
+
+> δV = 2πrh δr + πr2 δh
+
+> δr = 4 mm
+
+> δh = 1.2 mm
+
+> δV = 2π*15 cm*10 cm*0.4 cm + π*225 cm2*0.12 cm = 462 cm3
+
+## Summary
+
+A derivative is a measurement of the rate of change with respect to a certain variable. One of the most important derivatives is as follows:
+
+For f(x) = axn→ f’(x) = naxn-1.
+
+Important differentiation rules are the product rule, the quotient rule, and the chain rule. 
+
+Some functions contain multiple variables; you can take a derivative with respect to one of these variables. When differentiating with respect to one variable, the other variables in the function become constant.
+
+An important application of partial differentiation is small increment approximation, with which one can estimate small changes in a function. For some function f of x and y f(x,y), the change in f can be approximated using the following formula:
+
+> δf = (∂ f)/(∂ x)δx + (∂ f)/(∂ y)δy .
+
+
+## Glossary
+
+**Derivative:** a measurement of the **rate of change of a function with respect to a variable**. Derivatives are one of the fundamental tools for calculus.
+
+**Differentiation:** the operation of finding the derivative of a function.
+
+**Gradient/slope:** a number describing the direction and steepness of a function. For a straight line, the slope is equal to the distance travelled over the y-axis divided by the distance travelled over the x-axis. **For curved graphs, the slope at a certain point is equal to the derivative at that point.**
+
+**Tangent:** a straight line (or plane) that, at a certain point, **touches a curve but does not cross it at that point.**
+
+## Further Links to Videos and Additional Literature: 
+
+
+### Videos on Rules for Combined Functions: 
+
+**For an explanation and examples of the Product Rule:** 
+
+[![Product Rule](http://img.youtube.com/vi/B28EXpofKy4/0.jpg)](https://www.youtube.com/watch?v=B28EXpofKy4 "For an explanation and examples of the product rule")
+
+
+**For an explanation and examples of the quotient rule:**
+
+
+[![Quotient Rule](http://img.youtube.com/vi/O6M4O7zY5eA/0.jpg)](https://www.youtube.com/watch?v=O6M4O7zY5eA "For an explanation and examples of the quotient rule)
+
+
+**For an explanation on when and how to use the chain rule:**
+
 
-Or if video is preferred, these are also useful tools:
-
-https://www.youtube.com/watch?v=-_POEWfygmU&list=PL96AE8D9C68FEB902https://www.youtube.com/watch?v=-_POEWfygmU&list=PL96AE8D9C68FEB902
+[![Chain Rule](http://img.youtube.com/vi/H-ybCx8gt-8/0.jpg)](https://www.youtube.com/watch?v=H-ybCx8gt-8"For an explanation and examples of the chain rule)
 
-https://www.youtube.com/watch?v=EWVSxND_iWA&list=PLDesaqWTN6ESPaHy2QUKVaXNZuQNxkYQ_&index=2