Derivative of Parabola

We now understand the derivative of a parabola. It is commonly represented as d/dx (ax² + bx + c) or (ax² + bx + c)', and its value is 2ax + b. The quadratic function ax² + bx + c has a clearly defined derivative, indicating it is differentiable across its domain.

The key concepts are mentioned below:

Quadratic Function: A function of the form ax² + bx + c.

Power Rule: A rule for differentiating terms like ax².

Linear Function: The derivative of a quadratic function is a linear function.

The derivative of a parabola can be denoted as d/dx (ax² + bx + c) or (ax² + bx + c)'.

The formula we use to differentiate a quadratic function is: d/dx (ax² + bx + c) = 2ax + b

The formula applies to all x within the domain of a parabola.

We can derive the derivative of a parabola using proofs. To show this, we will use algebraic identities along with the rules of differentiation.

There are several methods we use to prove this, such as:

• By First Principle
   
• Using the Power Rule

We will now demonstrate that the differentiation of ax² + bx + c results in 2ax + b using the above-mentioned methods:

By First Principle

The derivative of ax² + bx + c can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative using the first principle, we will consider f(x) = ax² + bx + c. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = ax² + bx + c, we write f(x + h) = a(x + h)² + b(x + h) + c. Substituting these into the equation, f'(x) = limₕ→₀ [a(x + h)² + b(x + h) + c - (ax² + bx + c)] / h = limₕ→₀ [a(x² + 2xh + h²) + bx + bh + c - ax² - bx - c] / h = limₕ→₀ [2axh + ah² + bh] / h = limₕ→₀ [2ax + ah + b] As h approaches 0, the term ah vanishes, f'(x) = 2ax + b.

Using the Power Rule

To prove the differentiation of ax² + bx + c using the power rule, We differentiate each term separately: d/dx (ax²) = 2ax d/dx (bx) = b d/dx (c) = 0 Combining these gives: d/dx (ax² + bx + c) = 2ax + b.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. For a quadratic function, the first derivative is linear, and the second derivative is constant. To understand them better, consider a car where the position changes (function), the speed changes (first derivative), and the acceleration (second derivative) is constant. Higher-order derivatives simplify understanding of functions like ax² + bx + c.

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.

The second derivative is derived from the first derivative, which is denoted using f′′(x), and is constant for a parabola.

When the coefficient a is zero, the derivative reduces to a constant value b, representing a linear function. If b is also zero, the derivative becomes zero, indicating a constant function.

If a is positive, the parabola opens upwards, and its derivative increases. If a is negative, the parabola opens downwards, and its derivative decreases.

• Derivative: The derivative of a function indicates the rate of change of the function with respect to a variable.

• Quadratic Function: A second-degree polynomial function of the form ax² + bx + c.

• Power Rule: A fundamental rule in calculus for differentiating functions of the form xⁿ.

• First Derivative: The initial derivative of a function, representing the rate of change or slope of the function.

• Second Derivative: The derivative of the first derivative, indicating the rate of change of the rate of change, often related to acceleration.