(3) Differentiate y=(m−nx23+px43)a{\displaystyle y=\left(m-nx^{\frac {2}{3}}+{\dfrac {p}{x^{\frac {4}{3}}}}\right)^{a}}.
Let m−nx23+px−43=u{\displaystyle m-nx^{\frac {2}{3}}+px^{-{\frac {4}{3}}}=u}.
dudx=−23nx−13−43px−73{\displaystyle {\frac {du}{dx}}=-{\tfrac {2}{3}}nx^{-{\frac {1}{3}}}-{\tfrac {4}{3}}px^{-{\frac {7}{3}}}};
y=ua;dydu=aua−1{\displaystyle y=u^{a};\quad {\frac {dy}{du}}=au^{a-1}}.
dydx=dydu×dudx=−a(m−nx23+px43)a−1(23nx−13+43px−73).{\displaystyle {\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=1x3−a2{\displaystyle y={\dfrac {1}{\sqrt {x^{3}-a^{2}}}}}.
Let u=x3−a2{\displaystyle u=x^{3}-a^{2}}.
dudx=3x2;y=u−12;dydu=−12(x3−a2)−32{\displaystyle {\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}}}}.dydx=dydu×dudx=−3x22(x3−a2)3{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\times {\frac {du}{dx}}=-{\frac {3x^{2}}{2{\sqrt {(x^{3}-a^{2})^{3}}}}}}.
(5) Differentiate y=1−x1+x{\displaystyle y={\sqrt {\dfrac {1-x}{1+x}}}}.
Write this as y=(1−x)12(1+x)12{\displaystyle y={\dfrac {(1-x)^{\frac {1}{2}}}{(1+x)^{\frac {1}{2}}}}}.
dydx=(1+x)12d(1−x)12dx−(1−x)12d(1+x)12dx1+x{\displaystyle {\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)12(1+x)−12{\displaystyle y=(1-x)^{\frac {1}{2}}(1+x)^{-{\frac {1}{2}}}} and differentiate as a product.)