#math/calculus The **derivative** $f'(x_{0})$ is the limiting value of the difference quotient as $\Delta x$ becomes arbitrarily small. To calculate it from first principles, form $ \frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}, $ simplify for $\Delta x\neq 0$, and then let $\Delta x\to 0$. See [[1A6 - Derivative as a Limit]] for the formal definition. For $f(x)=ax^{2}+bx+c$, $ \frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x} =2ax_{0}+a\Delta x+b \longrightarrow 2ax_{0}+b. $ Thus $f'(x)=2ax+b$. The derivative of a quadratic is linear, the derivative of a linear function is constant, and the derivative of a constant is $0$.