The proof of the derivative of the quadratic functions is presented using the limit definition of the derivative .

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: \( c = 3 \)
\( f'(2) = 2 (-\dfrac{1}{2})(2) + 1 = -1 \)

More References and links

1. definition of the derivative
2. Proof of Derivative of \( \ln(x) \).
3. Proof of Derivative of \( cos(x) \).
4. Proof of Derivative of \( sin(x) \).
5. Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.
6. Derivatives of Sine (sin x) Functions. The derivative of sine functions are explored interactively.