Consider the line f(x)=mx+b, where m and b are constants. Show that f′(x)=m for all x. Interpret this result.

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 defined as the limit of the average rate of change of the function as the interval approaches zero: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
2. Step 2: Substitute the linear function f(x) = mx + b into the derivative definition. This gives us f'(x) = \lim_{h \to 0} \frac{(m(x+h) + b) - (mx + b)}{h}.
3. Step 3: Simplify the expression inside the limit. The terms b and -b cancel out, and we are left with f'(x) = \lim_{h \to 0} \frac{mx + mh - mx}{h}.
4. Step 4: Further simplify the expression. The terms mx and -mx cancel out, leaving f'(x) = \lim_{h \to 0} \frac{mh}{h}.
5. Step 5: Simplify the fraction \frac{mh}{h} to m, since h/h = 1 for h ≠ 0. Therefore, f'(x) = m for all x. This result shows that the slope of the line, m, is constant, and the derivative of a linear function is the slope of the line.