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Chapter 3: Problem 72

For the constant function \(f(x)=c,\) use the definition of the derivative to show that \(f^{\prime}(x)=0\).

Short Answer

Expert verified
Answer: The derivative of a constant function is 0. This is because a constant function has no rate of change, as its value remains the same for all input values. We can prove this by using the definition of the derivative and evaluating the limit, which results in a derivative of 0, signifying no change in the function.

Step by step solution

01

Write the derivative definition for the constant function

We start by writing the definition of the derivative for the constant function \(f(x) = c\): \(f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}\)
02

Replace \(f(x+h)\) and \(f(x)\) with the constant function

Now, we will replace the function values \(f(x+h)\) and \(f(x)\) with their constant values: \(f'(x) = \lim_{h\to 0} \frac{c - c}{h}\)
03

Simplify the numerator

Since the function is constant, we can simplify the difference in the numerator: \(f'(x) = \lim_{h\to 0} \frac{0}{h}\)
04

Evaluate the limit

Finally, we can evaluate the limit as \(h\) approaches 0: \(f'(x) = 0\) Therefore, for the constant function \(f(x) = c\), the derivative \(f'(x) = 0\).

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