Proof of the Derivative the Quadratic Function Using Definition of the Derivative

The definition of the derivative \( f' \) of a function \( f \) is given by the limit
\[ f'(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h} \]
Let \( f \) be a quadratic function of the form: \( f(x) = a x^2 + bx + c \) and write the derivative of \( f \) as follows
\( f'(x) = \lim_{h \to 0} \dfrac{ a (x+h)^2 + b(x+h) + c - (a x^2 + bx + c)}{h} \)

Expand the terms \( a (x+h)^2 \) and \( b(x+h) \) in the numerator
\( f'(x) = \lim_{h \to 0} \dfrac{ a x^2 + a h^2 + 2 a x h + b x + bh + c - a x^2 - bx - c)}{h} \)

Simplify the numerator
\( f'(x) = \lim_{h \to 0} \dfrac{ a h^2 + 2 a x h + bh )}{h} \)

Divide numerator and denomianator by \( h \)
\( f'(x) = \lim_{h \to 0} a h + 2 a x + b \)

Evaluate the limit to obtain the derivative of the quadratic function as
\( f'(x) = 2 a x + b \)

Examples with Solutions

Part A
Find the derivatives of the quadratic functions given by
a) \( f(x) = 4x^2 - x + 1 \)
b) \( g(x) = - x^2 - 1 \)
c) \( h(x) = 0.1 x^2 - \dfrac {x}{2} - 100 \)
d) \( f(x) = - \dfrac { 3 x^2}{7} - 0.2 x + 7\)

Part B
Let \( f(x) = a x^2 + b x + c \).
Find \( f'(2) \) given that \( f(2) = 3 \), \( f'(0) = 1 \) and \( f'(-1) = 2 \)

Solutions to the Above Examples

Part A
a) \( f'(x) = 8 x - 1 \)
b) \( g'(x) = - 2 x \)
c) \( h'(x) = 0.2 x - \dfrac {1}{2} \)
d) \( f'(x) = -\dfrac {6 x}{7} - 0.2 \)

Part B
The given function \( f \) is a quadratic function, hence
\( f'(x) = 2 a x + b \)
Given \( f'(0) = 1 \), substitute to obtain the equation: \( 2 a (0) + b = 1 \)
Solve for \( b \) to obtain: \( b = 1 \)
Given \( f'(-1) = 2 \), substitute to obtain the equation: \( 2 a (-1) + 1 = 2 \)
Solve for \( a \) to obtain: \( a = -\dfrac{1}{2} \)
Given \( f(2) = 3 \), substitute to obtain the equation: \( f(2) = -\dfrac{1}{2} (2)^2 + (1) (2) + c = 3\)
Solve for \( c \) to obatain: \( c = 3 \)
\( f'(2) = 2 (-\dfrac{1}{2})(2) + 1 = -1 \)