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LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES2 Marks
Question
Find the derivative of $\frac{{{x^n} - {a^n}}}{{x - a}}$ for some constant a.
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Answer

Here $f(x)\frac{{{x^n} - {a^n}}}{{x - a}}$
$\therefore \;f(x) = \frac{d}{{dx}}\left[ {\frac{{{x^n} - {a^n}}}{{x - a}}} \right]$
$= \frac{{(x - a)\frac{d}{{dx}}({x^n} - {a^n}) - ({x^n} - {a^n})\frac{d}{{dx}}(x - a)}}{{{{(x - a)}^2}}}$
$ = \frac{{(x - a) \times n{x^{n - 1}} - ({x^n} - {a^n}) \times 1}}{{{{(x - a)}^2}}}$
$= \frac{{n{x^n} - an{x^{n - 1}} - {x^n} + {a^n}}}{{{{(x - a)}^2}}}$

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