VCE Mathematical Methods/Differentiation from First Principles
Theory
editFormula
editGiven a function f, the rule of the derivative (sometimes called the "gradient") function is defined as .
Method
editRemember 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
editQuestion 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)