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Chapter 2: Problem 14

Find the derivative by the limit process. \(f(x)=9-\frac{1}{2} x\)

Short Answer

Expert verified
The derivative of the function \(f(x) = 9 - \frac{1}{2}x\) is \(f'(x) = -\frac{1}{2}\).

Step by step solution

01

Understand the problem

The problem is to find the derivative of the function \(f(x) = 9 - \frac{1}{2}x\). This demands us to apply the limit definition of the derivative.
02

Apply the limit definition of the derivative

To do so, find the limit as \(h\) approaches zero of: \[\frac{f(x+h)-f(x)}{h}\] Where \(f(x+h)\) involves replacing every instance of \(x\) in the original function with \((x+h)\), and \(f(x)\) is the original function.
03

Performing the calculations

\[f(x + h) = 9 - \frac{1}{2}(x + h)\] Next, replace \(f(x + h)\) and \(f(x)\) in the definition with obtained and original function respectively: \[\frac{f(x + h) - f(x)}{h} = \frac{(9 - \frac{1}{2}(x + h)) - (9 - \frac{1}{2}x)}{h}\] Simplify the above equation to find the limit as \(h\) approaches zero.
04

Simplification

After simplification, the equation becomes: \[\lim_{h\to0} -\frac{1}{2}\] Note: This simplifies directly because \(-\frac{1}{2}\) does not depend on \(h\). This concludes that \(f'(x) = -\frac{1}{2}\).

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