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Find the derivative of the following functions form first prin

Question

Find the derivative of the following functions form first principle:
$(- x)^{- 1}$

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Lucas Discussed

Find the derivative of the following functions form first principle:

(- x)^{- 1}

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Text solutionVerified

Let

f (x) = (- x)^{- 1} = \frac{- 1}{x}

Thus using first principle

f^{'} (x) = x \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

x \to 0 lim \frac{1}{h} [\frac{- 1}{x + h} - (\frac{- 1}{x})]

x \to 0 lim \frac{1}{h} [\frac{- 1}{x + h} + \frac{1}{x}]

x \to 0 lim \frac{1}{h} [\frac{- x + ( x + h )}{x ( x + h )}]

= h \to 0 lim \frac{1}{h} [\frac{- x + x + h}{x ( x + h )}]

= x \to 0 lim \frac{1}{h} [\frac{h}{x ( x + h )}]

= x \to 0 lim \frac{1}{x ( x + h )} = \frac{1}{x ^{2}}

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Question Text	Find the derivative of the following functions form first principle: $(- x)^{- 1}$
Answer Type	Text solution:1
Upvotes	150