Differentiation from first principles
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Differential Calculus
Equations are omitted for technical reasons - download the original pdf
The differential calculus provides a mathematical means of arriving at the gradient of a tangent to a function. Firstly, we calculate the gradient of a cord joining two points on the graph of the function: [Diagram goes here - download the original pdf to see it.] The gradient of the cord joining [Equation goes here - download the original pdf to see it.] where [Equation goes here - download the original pdf to see it.] are small increments in x and y respectively, is [Equation goes here - download the original pdf to see it.] Sometimes the symbol h (or some other letter) is used for the small increment; in this case the gradient is given by [Equation goes here - download the original pdf to see it.] It is the same idea, just with a different letter. The differential calculus arises from the following idea. As [Equation goes here - download the original pdf to see it.] gets smaller and smaller, the cord joining [Equation goes here - download the original pdf to see it.] gets closer and closer to the tangent at x. The following diagram should help you to understand this. [Diagram goes here - download the original pdf to see it.] If becomes "infinitesimally small", then the cord becomes the tangent. Of course, there is a problem with the expression "infinitesimally small" - does it really mean anything. Actually, what we really mean is that as [Equation goes here - download the original pdf to see it.] (as [Equation goes here - download the original pdf to see it.] gets closer and closer to 0) then the gradient of the cord gets closer and closer to the gradient of the tangent. This is really a statement about limits so eventually we have to recast the whole theory of the differential calculus in terms of limits. However, for now we will pretend that the phrase "infinitesimally small" does mean something, and also that in practice is amounts to putting [Equation goes here - download the original pdf to see it.] at some point in the argument. We denote the gradient of the tangent by [Equation goes here - download the original pdf to see it.] (There are many different symbols used for the derivative.) Here dy and dx in [Equation goes here - download the original pdf to see it.] mean the increments in y and x respectively along the tangent at a given point a to the graph of [symbol] [Diagram goes here - download the original pdf to see it.] Hence [Equation goes here - download the original pdf to see it.]or Equation goes here - download the original pdf to see it.] It is this definition that we call the definition of the derivative from first principles. Note here that and h are totally interchangeable. They mean the same thing, but we give both forms of the same equation because you see both in the literature. When you are asked (in this context) to find a derivative from first principles, it is with this equation (or equivalent) that you start. Example (1) The derivative of [Equation goes here - download the original pdf to see it.] from first principles is [Example goes here - download the original pdf to see it.] Let us annotate this solution. At the line [Equation goes here - download the original pdf to see it.] the function [Equation goes here - download the original pdf to see it.] is substitute for y. This makes the next line strictly redundant. The next line [Equation goes here - download the original pdf to see it.] is the definition of the derivative from first principles. This is where we start, and actually the first line could be omitted. The function is [Equation goes here - download the original pdf to see it.] so we square [symbol] and x to obtain the next line [Equation goes here - download the original pdf to see it.] This has removed the general function f and replaced it by the specific function [symbol]. Now follows some algebra in which the term [Equation goes here - download the original pdf to see it.] is unsquared. This makes it possible to cancel out some terms. Then is a factor of both the numerator (top) and denominator (bottom) of the fraction, so it is cancelled through. At this point we have [Equation goes here - download the original pdf to see it.] Here we are allowed by our assumption that taking the limit is equivalent to putting [Equation goes here - download the original pdf to see it.] so when we "take the limit" we merely substitute [Equation goes here - download the original pdf to see it.] and scratch out the [Equation goes here - download the original pdf to see it.] part. Note, this step is only allowed if there is no zero in the denominator (the bottom) of a fraction, for otherwise it is equivalent to dividing by zero, and that is not allowed - it leads to contradictions. Example (2) Find the derivative of [Equation goes here - download the original pdf to see it.] from first principles. [Example goes here - download the original pdf to see it.] Note that in the expression [Equation goes here - download the original pdf to see it.] we are allowed to substitute [symbol] because the result does not lead to a zero on the bottom of the fraction. However, it is because of the potential threat of dividing by zeros that strictly speaking the whole calculus should be recast in terms of a theory of limits. But this is left for a later chapter.
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Contents of
Differentiation from first principles
1 Differentiation from First Principles Prerequisites
2 Tangent, Gradient and Rate of Change
3 Differential Calculus
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