Introduction / Summary of Differentiationby Hootenanny Tags: differentiation, introduction, summary 

#1
Oct2306, 09:42 AM

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As the title suggests this thread is intended as a summary of differentiation, it is by no means an attempt at an exhaustive discussion of the topic. A detailed knowledge of limits is not required but is useful, it is assumed that the reader has some knowledge of limits and how to implement them. As always I welcome comments and corrections either here of via PM.
The Definition of a Derivative The derivative ([itex]f'(x)[/itex]) of a function [itex]f(x)[/itex] with respect to [itex]x[/itex] is given by the limit; [tex]f'(x)=\lim_{h\to0}\;\frac{f(x+h)f(x)}{h}[/tex] Provided the limit exists. If the limit exist at a point x0 we say that the function is differentiable at this point. As can be seen from the plots below, as [itex]h[/itex] approaches zero, the secants of the curve tend to the tangent at a point x, this gives the gradient of the curve at point x (provided the function is differentiable at point x). Note however, that in the above limit we cannot simply set [itex]h[/itex] as zero directly as our limit would [itex]\to\pm\infty[/itex] and thus, the limit would not exist. We must therefore manipulate the limit into a form where this does not occur. Images taken from Wikipedia ExampleFinding the Gradient and Tangent to a Curve at a Point We already now that the derivative of a function gives the gradient of a curve at any point that is differentiable. This allows us to find the gradient of the function at any point x where the curve is differentiable. This then allows us to find the equation of the tangent to the curve at any point x where the function is differentiable. Example 



#2
Oct2306, 12:48 PM

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Rules of Differentiation
Using the definition of a derivative every time we wish to differentiate a function becomes tedious after a while. It is therefore desirable to derive a series of rules which allows us to differentiate a variety of functions without having to refer to the above definition. 1. Derivative of a Constant Function 



#3
Oct2806, 12:12 PM

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5. Product Rule 



#4
Oct2906, 03:38 AM

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Introduction / Summary of Differentiation6. The Quotient Rule 



#5
Nov906, 01:05 PM

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7. Power Rule for Negative Integers 



#6
Nov906, 05:10 PM

P: 44

Can you continue proving the Power rule for all real numbers using logarithmic differentiation.




#7
Nov1006, 04:54 AM

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Thank you for your comments. I am planning to discuss the definition and differentiation of logarithmic and exponential functions later in the tutorial. Prior to this I will consider the chain rule, implicit differentiation and differentiation of trigonometric functions. 



#8
Nov1006, 02:15 PM

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8. The Chain Rule 



#9
Nov2206, 09:35 AM

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A correction in post number four. The formula should read;
[tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\left (\frac{u}{v}\right)=\frac{v\frac{du}{dx}u\frac{dv}{dx}}{v^2}}}}[/tex] As an aside it can also be expressed, perhaps more succinctly, thus; [tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\left (\frac{u}{v}\right)=\frac{v\cdot u'  u\cdot v'}{v^2}}}}[/tex] 



#10
Dec506, 01:01 PM

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Implicit Differentiation
Up until this point we have always considered functions which can be written explicitly in the form of [itex]y=ax^n + bx^{n1}...[/itex], these are known an explicit functions. However, some functions or equations can not be written explicitly, instead they must be define in terms of a relationship between two variables, these are known as implicit functions. An example of an implicit function would be that of a unit circle; [itex]x^2 + y^2 =1[/itex]. On occasions it may be possible to write such equations in an explicit form, however, it is often preferable to differentiate implicit functions implicitly as they are. When we differentiate implicitly we differentiate the implicit function with respect to the desired variable (x for example); we then treat other variables (y for example) as unknown functions of x and differentiate them accordingly using the chain rule. There is no formula to learn here (except the chain rule obviously), it is simply a technique you must practise. Example; 



#11
Dec606, 05:18 PM

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A typo in my previous post. The penultimate line in the example should read as follows;
[tex] 2x + {\color{red}2}\cdot y\cdot\frac{dy}{dx} = 0[/tex] And therefore the final line will read; [tex]\therefore \;\;\; m = \frac{dy}{dx} = \frac{x}{y}[/tex] Apologies for any confusion. 



#12
Dec806, 05:22 AM

P: 1,142

In the example given for the quotient rule the answer should be, I hope:
[tex]\frac{d}{dx}\left(\frac{x^2+1}{x^21}\right)[/tex] [tex]\Rightarrow \frac{d}{dx}\frac{x^2+1}{x^21} = \frac{2x\cdot(x^21)  2x\cdot(x^2+1)}{(x^21)^2} = \frac{4x}{(x^21)^2}[/tex] You have no idea how long I puzzled over your answer before I realised that it must be wrong. 



#13
Dec806, 10:31 AM

P: 75

firstly as setting h = 0 would make numerator as well as denominator zero, we've no need for the intuitive idea that the expression becomes infinity. secondly the bolded limit might be better replaced with expression and the arrow removed, as setting to zero, rather than limiting, makes it no longer a limit and means ideas like +/ infinity are out of place an explanation of why the differential is undefined at h = 0 and the need for a limit as a consequence of this might be better 



#14
Dec1206, 06:22 AM

P: 75

i saw a reply to my point immediately above, but then it was deleted.
? 



#15
Dec1206, 08:30 AM

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P.S. I removed my previous post as I felt that I didn't give your post sufficient attention. 



#16
Dec1206, 09:25 AM

P: 75





#17
Dec1206, 10:16 AM

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Differentiation of Trigonometric Functions
Trigonometric functions has many important applications in science, especially physics. Therefore, it is important that we are able to differentiate equations involving trigonometric functions. Here I intend to illustrate how to differentiate trigonometric functions from first principles and present a series of 'rules' which I would recommend committing to memory if you are going to be dealing with trigonometric functions on a regular basis. 1. Derivative of Sine 



#18
Dec1806, 09:04 AM

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2. Derivative of Cosine 


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