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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.$ )
$$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\lim _{h \rightarrow 0} \frac{x-(x+h)}{h x(x+h)}=\lim _{h \rightarrow 0} \frac{-h}{h x(x+h)}=\lim _{h \rightarrow 0} \frac{-1}{x(x+h)}=-\frac{1}{x^{2}}$$
01:01
Frank L.
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 1
Derivatives of Polynomials and Exponential Functions
Derivatives
Differentiation
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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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