02) Differentiation: Basic Rules

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 third rule, is known as the Product Rule, since multiplication can be considered as the "Product of Multiple Values". For example, the product of 3x and 2y is 6xy. Tangentially, the Product Rule is used on the product of two or more functions:

The fourth rule, is known as the Quotient Rule, since a Quotient is defined as "a result obtained by dividing one quantity by another". In fact, this rule can be formed by the first and second rule.



The fifth rule, is known as the Chain Rule, due to the fact the original variation of it is used to connect multiple derivatives of different variables in a chain-like manner.



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