Page:Calculus Made Easy.pdf/173

(10) $y={\dfrac {1}{\log _{\epsilon }x}}$.

${\frac {dy}{dx}}={\frac {\log _{\epsilon }x\times 0-1\times {\dfrac {1}{x}}}{\log _{\epsilon }^{2}x}}=-{\frac {1}{x\log _{\epsilon }^{2}x}}.$

(11) $y={\sqrt[{3}]{\log _{\epsilon }x}}=(\log _{\epsilon }x)^{\frac {1}{3}}$. Let $y=z^{\frac {1}{3}}$.

${\frac {dy}{dz}}={\frac {1}{3}}z^{-{\frac {2}{3}}};\quad {\frac {dz}{dx}}={\frac {1}{x}};\quad {\frac {dy}{dx}}={\frac {1}{3x{\sqrt[{3}]{\log _{\epsilon }^{2}x}}}}.$

(12) $y=\left({\dfrac {1}{a^{x}}}\right)^{ax}$.

${\begin{aligned}\log _{\epsilon }y&=ax(\log _{\epsilon }1-\log _{\epsilon }a^{x})=-ax\log _{\epsilon }a^{x}.\\{\frac {1}{y}}\,{\frac {dy}{dx}}&=-ax\times a^{x}\log _{\epsilon }a-a\log _{\epsilon }a^{x}.\\and\quad \quad \quad {\frac {dy}{dx}}&=-\left({\frac {1}{a^{x}}}\right)^{ax}(x\times a^{x+1}\log _{\epsilon }a+a\log _{\epsilon }a^{x}).\end{aligned}}$

Try now the following exercises.

 

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Exercises XII. (See page 260 for Answers.)

(1) Differentiate $y=b(\epsilon ^{ax}-\epsilon ^{-ax})$.

(2) Find the differential coefficient with respect to $t$ of the expression $u=at^{2}+2\log _{\epsilon }t$.

(3) if $y=n^{t}$, find ${\dfrac {d(\log _{\epsilon }y)}{dt}}$.

(4) Show that if $y={\dfrac {1}{b}}\cdot {\dfrac {a^{bx}}{\log _{\epsilon }a}}$ ; ${\dfrac {dy}{dx}}=a^{bx}$.

(5) If $w=pv^{n}$, find ${\dfrac {dw}{dv}}$.