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LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES1 Mark
Question
Find the derivative of $f ( x ) = \frac { 1 } { x }$
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Answer

We have, $f ( x ) = \frac { 1 } { x }$
By using the first principle,
$f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h) - f ( x ) } { h }$
$\therefore \quad {f^\prime }(x) = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{x + h}} - \frac{1}{x}}}{h}$ $\left[ {\begin{array}{*{20}{c}} {\because f(x) = \frac{1}{x}} \\ {\therefore f(x + h) = \frac{1}{{x + h}}} \end{array}} \right]$
$ = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left[ {\frac{{x - (x + h)}}{{x(x + h)}}} \right] = \mathop {\lim }\limits_{h \to 0} \frac{1}{h}\left[ {\frac{{ - h}}{{x(x + h)}}} \right]$
$ = \mathop {\lim }\limits_{h \to 0} \left[ {\frac{{ - 1}}{{x(x + h)}}} \right] = \frac{{ - 1}}{{{x^2}}}$

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