1. The problem is to find the derivative of a quadratic equation and show the steps for its derivation. 2. Consider the quadratic function $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. 3. The derivative of $f(x)$, denoted $f'(x)$ or $\frac{df}{dx}$, measures the rate of change of the function with respect to $x$. 4. Using the power rule of differentiation, which states that $\frac{d}{dx}[x^n] = nx^{n-1}$, apply it to each term: - For $ax^2$, the derivative is $2ax^{2-1} = 2ax$. - For $bx$, the derivative is $b$ because $\frac{d}{dx}[x] = 1$. - For $c$, a constant, the derivative is $0$. 5. Combine these results to write the full derivative: $$ f'(x) = 2ax + b $$ 6. This means the derivative of the quadratic function $ax^2 + bx + c$ is a linear function $2ax + b$, which describes the slope of the tangent line to the curve at any point $x$.