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Problem 65 Easy Difficulty

Use the definition of a derivative to show that if $ f(x) = 1/x, $ then $ f'(x) = -1/x^2. $ (This proves the Power Rule for the case $ n = -1.$ )

Answer

$$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\lim _{h \rightarrow 0} \frac{x-(x+h)}{h x(x+h)}=\lim _{h \rightarrow 0} \frac{-h}{h x(x+h)}=\lim _{h \rightarrow 0} \frac{-1}{x(x+h)}=-\frac{1}{x^{2}}$$

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Video Transcript

Hey, it's clear, So enumerated here. So we have up of access equal Thio one over X. Yeah, if the door If it is, it's equal to limit. As each approaches Ciro one over X Plus H minus one over X over each becomes equal to the limit knows each approaches Ciro for a negative H over X X plus H over each. This is equal to limit as each approaches. So for negative one over X terms X plus H, which is equal to negative one over X square.

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