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

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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} \(\frac{f(x+h) - f(x)}{h}\).

Step 2: Substitute the linear function f(x) = mx + b into the derivative definition. This gives us f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{(m(x+h) + b) - (mx + b)}{h}\).

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} \(\frac{mx + mh - mx}{h}\).

Step 4: Further simplify the expression. The terms mx and -mx cancel out, leaving f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{mh}{h}\).

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For a linear function like f(x) = mx + b, the derivative represents the slope of the line, which is constant.

Linear Functions

A linear function is a polynomial function of degree one, represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line, and its slope (m) indicates how steep the line is. Since the slope is constant, the derivative of a linear function is the same for all values of x.

Interpretation of the Derivative

The derivative can be interpreted as the instantaneous rate of change of a function at a given point. In the context of the linear function f(x) = mx + b, the result f′(x) = m indicates that the rate of change is constant across all x-values. This means that for every unit increase in x, the function f(x) increases by m units, reflecting the uniform behavior of linear functions.