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 f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \, .

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  h = 0 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  f(x+h) and  f(x) , then simplify to eliminate the fraction, and only then substitute  h = 0 . This is called differentiation from first principles.

For example:

Let  f: \mathbb{R} \to \mathbb{R}, f(x) = 2x

Let us differentiate f from first principles.


\begin{align}
 f'(x)  & = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \\
       & = \lim_{h \to 0} \frac{2(x+h) - 2x}{h} \\
       & = \lim_{h \to 0} \frac{2x + 2h - 2x}{h} \\
       & = \lim_{h \to 0} \frac{2h}{h} \\
       & = \lim_{h \to 0} 2 \\
       & = 2 \\
\end{align}
.
Therefore, we can define the gradient function as  f': \mathbb{R} \to \mathbb{R}, f'(x) = 2

Exercises[edit]

Question One
Differentiate the following functions from first principles.
(a)  f(x) = 4x
(b)  f(x) = 7x
(c)  f(x) = 2x + 1
(d)  f(x) = 3x + 3

Question Two
Differentiate the following functions from first principles.
(a)  g(x) = x^2
(b)  f(x) = 5x^2
(c)  f(x) = 2x^2 + 3
(d)  f(x) = (x+3)(x+4)