VCE Mathematical Methods/Differentiation from First Principles

Theory

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Formula

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Given a function f, the rule of the derivative (sometimes called the "gradient") function is defined as  .

Method

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Remember that in order to evaluate a limit, we usually substitute the value given into the expression. However, with the above formula, substituting   will result in a division by zero, which is mathematically impossible. Therefore,in order to make use of this formula, you need to substitute the rules   and  , then simplify to eliminate the fraction, and only then substitute  . This is called differentiation from first principles.

For example:

Let  

Let us differentiate f from first principles.

 .
Therefore, we can define the gradient function as  

Exercises

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Question One
Differentiate the following functions from first principles.
(a)  
(b)  
(c)  
(d)  

Question Two
Differentiate the following functions from first principles.
(a)  
(b)  
(c)  
(d)