Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Verified step by step guidance

1. Step 1: Recall the definition of the derivative. The derivative of a function f(x) at a point x is given by the limit: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
2. Step 2: Identify the function f(x) = ax^2 + bx + c. We need to find f(x+h) by substituting x with (x+h) in the function: f(x+h) = a(x+h)^2 + b(x+h) + c.
3. Step 3: Expand f(x+h). Use the distributive property to expand: a(x+h)^2 = a(x^2 + 2xh + h^2), and b(x+h) = bx + bh. So, f(x+h) = ax^2 + 2axh + ah^2 + bx + bh + c.
4. Step 4: Substitute f(x) and f(x+h) into the derivative definition: \( f'(x) = \lim_{h \to 0} \frac{(ax^2 + 2axh + ah^2 + bx + bh + c) - (ax^2 + bx + c)}{h} \).
5. Step 5: Simplify the expression inside the limit. Cancel out the terms ax^2, bx, and c, and then divide each remaining term by h: \( f'(x) = \lim_{h \to 0} \frac{2axh + ah^2 + bh}{h} = \lim_{h \to 0} (2ax + ah + b) \). Finally, evaluate the limit as h approaches 0.