2) Fundamentals Exponentials

 




Thus a rational number with an irrational as an exponent is rational.



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04) Differentiation: Basic Rules Theory

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 The first rule, the Power Rule, is the rule used upon functions of x that are bases for non-variable values. The second rule, the Reciprocal Rule, is used on the reciprocal of certain functions or identities, however this rule can be derived (aka found from) the Power Rule. A reciprocal is merely 1 divided by a function, this can be rewritten as f(x)^(-1), thus the function becomes the base of the power of "-1". The third rule, the product rule, is the rule used on the products of multiple functions and identities. This rule is based off the initial theory used to formulate differentiation, the use of the difference of hypothetical infinitesimally small differences. As the derivative is the change, we only consider the two blue boxes of uv' and u'v as a result of the change, we can ignore the u'v' since both values are small, we can consider it negligible. The fourth rule, the Quotient Rule, derivable from the Product Rule The fifth rule, the Chain rule, a si...
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02) Differentiation: Basic Rules

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There are many "Rules" of differentiation, but in a sense these are shorthand which skip over the complicated proof that show why they work. Although, at the level of IGCSE these proofs are unnecessary to be placed in each question they are used, and can be assumed to be true when used, enough paper is used as it is. However, I do encourage to look and attempt to understand the proof, as it is a useful method for remembering the how and when to use them. The proof of each can be seen through the link to the eventual page for it:  Here The first rule, is known as the Power Rule, since it deals with functions of x by the power of a non-variable value. For example: f(x)^n. Can be differentiated by reducing the power by 1, and multiplying by the power and the derivative of f(x). The second rule, is known as the Reciprocal Rule, since the reciprocal of f(x) is 1/f(x) or (f(x))^(-1). In fact, this rule is formed by the Power Rule, however remembering this as it is, is fine, The thi...
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