Page:Calculus Made Easy.pdf/274

ANSWERS.

 

Exercises I. (p. 25.)

(1) ${\dfrac {dy}{dx}}=13x^{12}$.

(2) ${\dfrac {dy}{dx}}=-{\dfrac {3}{2}}x^{-{\frac {5}{2}}}$.

(3) ${\dfrac {dy}{dx}}=2ax^{(2a-1)}$.

(4) ${\dfrac {du}{dt}}=2.4t^{1.4}$.

(5) ${\dfrac {dz}{du}}={\dfrac {1}{3}}u^{-{\frac {2}{3}}}$.

(6) ${\dfrac {dy}{dx}}=-{\dfrac {5}{3}}x^{-{\frac {8}{3}}}$.

(7) ${\dfrac {du}{dx}}=-{\dfrac {8}{5}}x^{-{\frac {13}{5}}}$.

(8) ${\dfrac {dy}{dx}}=2ax^{a-1}$.

(9) ${\dfrac {dy}{dx}}={\dfrac {3}{q}}x^{\frac {3-q}{q}}$.

(10) ${\dfrac {dy}{dx}}=-{\dfrac {m}{n}}x^{-{\frac {m+n}{n}}}$.

 

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Exercises II. (p. 33.)

(1) ${\dfrac {dy}{dx}}=3ax^{2}$.

(2) ${\dfrac {dy}{dx}}=13\times {\frac {3}{2}}x^{\frac {1}{2}}$.

(3) ${\dfrac {dy}{dx}}=6x^{-{\frac {1}{2}}}$.

(4) ${\dfrac {dy}{dx}}={\dfrac {1}{2}}c^{\frac {1}{2}}x^{-{\frac {1}{2}}}$.

(5) ${\dfrac {du}{dz}}={\dfrac {an}{c}}z^{n-1}$.

(6) ${\dfrac {dy}{dt}}=2.36t$.

(7) ${\dfrac {dl_{t}}{dt}}=0.000012\times l_{0}$.

(8) ${\dfrac {dC}{dV}}=abV^{b-1}$, $0.98$, $3.00$ and $7.47$ candle power per volt respectively.

(9) ${\dfrac {dn}{dD}}=-{\dfrac {1}{LD^{2}}}{\sqrt {\dfrac {gT}{\pi \sigma }}},{\dfrac {dn}{dL}}=-{\dfrac {1}{DL^{2}}}{\sqrt {\dfrac {gT}{\pi \sigma }}}$.
$\;\quad {\dfrac {dn}{d\sigma }}=-{\dfrac {1}{2DL}}{\sqrt {\dfrac {gT}{\pi \sigma ^{3}}}},{\dfrac {dn}{dT}}={\dfrac {1}{2DL}}{\sqrt {\dfrac {g}{\pi \sigma T}}}$.