The first derivative of a quadratic function f(x) = ax² + bx + c is f'(x) = 2ax + b.
Understanding the Derivative of a Quadratic Function
Let's break down why the derivative of a quadratic function takes this form.
Quadratic Function Form
A quadratic function is generally expressed as:
- f(x) = ax² + bx + c
Where:
- 'a', 'b', and 'c' are constants.
- 'x' is the variable.
Applying the Power Rule of Differentiation
To find the derivative, we use the power rule of differentiation, which states:
- If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
Applying this rule to each term in the quadratic function:
- Derivative of ax²:
- Using the power rule: 2 * ax²⁻¹ = 2ax
- Derivative of bx:
- This is equivalent to bx¹, so applying the power rule: 1 * bx¹⁻¹ = b
- Derivative of c:
- The derivative of a constant is 0.
The Resulting First Derivative
Combining the derivatives of each term, we get:
- f'(x) = 2ax + b + 0
- f'(x) = 2ax + b
Practical Insights
- Slope: The first derivative f'(x) of a quadratic function gives the slope of the tangent line to the curve at any given point x.
- Linear Nature: As noted in the provided reference, the first derivative is a linear function (2ax + b). This is because the original quadratic function's degree is 2.
- Finding Minimum/Maximum: The derivative is essential in finding the minimum or maximum value of a quadratic function. The point where the derivative equals zero represents the vertex of the parabola.
Example:
Let's take the quadratic function: f(x) = 3x² + 5x + 2
The derivative, based on our formula f'(x) = 2ax + b, would be f'(x) = 2(3)x + 5 = 6x + 5
Key Takeaway
The derivative of a quadratic function is always a linear function, a crucial piece of information for understanding its behavior. As mentioned in the reference, the discriminant (D = b² - 4ac) is a completely separate calculation and should not be confused with the first derivative (f'(x) = 2ax + b).
