How to differentiate a quadratic equation?

To differentiate a quadratic equation, apply the power rule and constant multiple rule.

Let's clarify the process of differentiating a quadratic equation. A quadratic equation generally takes the form:

f(x) = ax² + bx + c

Where a, b, and c are constants. To find the derivative, f'(x), we apply the power rule (d/dx (xⁿ) = nx^n-1) and the constant multiple rule (d/dx (cf(x)) = c d/dx f(x)). We also know the derivative of a constant is zero.

Here's how to do it step-by-step:

1. Differentiate the first term (ax²):

   • Apply the power rule: d/dx (x²) = 2x¹ = 2x
   • Apply the constant multiple rule: d/dx (ax²) = a * 2x = 2ax

2. Differentiate the second term (bx):

   • Remember that x is x¹
   • Apply the power rule: d/dx (x¹) = 1x⁰ = 1 (since anything to the zero power equals 1 - from the reference material)
   • Apply the constant multiple rule: d/dx (bx) = b * 1 = b

3. Differentiate the third term (c):

   • The derivative of a constant is zero: d/dx (c) = 0

4. Combine the results:

f'(x) = 2ax + b + 0 = 2ax + b

Example:

Let's say we have the quadratic equation:

f(t) = -3t² + 2t + 4

Following the steps:

1. Derivative of -3t² is -6t
2. Derivative of 2t is 2
3. Derivative of 4 is 0 (From the reference: "And the derivative of a constant is 4 simplifying I get - 6 T 1st + 2 remember t to the 0. Equals. 1")

Therefore:

f'(t) = -6t + 2