Page:Calculus Made Easy.pdf/89

(3) Differentiate $y=\left(m-nx^{\frac {2}{3}}+{\dfrac {p}{x^{\frac {4}{3}}}}\right)^{a}$.

Let $m-nx^{\frac {2}{3}}+px^{-{\frac {4}{3}}}=u$.

${\frac {du}{dx}}=-{\tfrac {2}{3}}nx^{-{\frac {1}{3}}}-{\tfrac {4}{3}}px^{-{\frac {7}{3}}}$;

$y=u^{a};\quad {\frac {dy}{du}}=au^{a-1}$.

${\frac {dy}{dx}}={\frac {dy}{du}}\times {\frac {du}{dx}}=-a\left(m-nx^{\frac {2}{3}}+{\frac {p}{x^{\frac {4}{3}}}}\right)^{a-1}({\tfrac {2}{3}}nx^{-{\frac {1}{3}}}+{\tfrac {4}{3}}px^{-{\frac {7}{3}}}).$

(4) Differentiate $y={\dfrac {1}{\sqrt {x^{3}-a^{2}}}}$.

Let $u=x^{3}-a^{2}$.

${\frac {du}{dx}}=3x^{2};\quad y=u^{-{\frac {1}{2}}};\quad {\frac {dy}{du}}=-{\frac {1}{2}}(x^{3}-a^{2})^{-{\frac {3}{2}}}$.
${\frac {dy}{dx}}={\frac {dy}{du}}\times {\frac {du}{dx}}=-{\frac {3x^{2}}{2{\sqrt {(x^{3}-a^{2})^{3}}}}}$.

(5) Differentiate $y={\sqrt {\dfrac {1-x}{1+x}}}$.

Write this as $y={\dfrac {(1-x)^{\frac {1}{2}}}{(1+x)^{\frac {1}{2}}}}$.

${\frac {dy}{dx}}={\frac {(1+x)^{\frac {1}{2}}\,{\dfrac {d(1-x)^{\frac {1}{2}}}{dx}}-(1-x)^{\frac {1}{2}}\,{\dfrac {d(1+x)^{\frac {1}{2}}}{dx}}}{1+x}}$.

(We may also write $y=(1-x)^{\frac {1}{2}}(1+x)^{-{\frac {1}{2}}}$ and differentiate as a product.)