VCE Mathematical Methods/Differentiation from First Principles

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Theory[edit]

Formula[edit]

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

Method[edit]

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[edit]

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)