(3) Differentiate y = ( m − n x 2 3 + p x 4 3 ) a {\displaystyle y=\left(m-nx^{\frac {2}{3}}+{\dfrac {p}{x^{\frac {4}{3}}}}\right)^{a}} .
Let m − n x 2 3 + p x − 4 3 = u {\displaystyle m-nx^{\frac {2}{3}}+px^{-{\frac {4}{3}}}=u} .
d u d x = − 2 3 n x − 1 3 − 4 3 p x − 7 3 {\displaystyle {\frac {du}{dx}}=-{\tfrac {2}{3}}nx^{-{\frac {1}{3}}}-{\tfrac {4}{3}}px^{-{\frac {7}{3}}}} ;
y = u a ; d y d u = a u a − 1 {\displaystyle y=u^{a};\quad {\frac {dy}{du}}=au^{a-1}} .
d y d x = d y d u × d u d x = − a ( m − n x 2 3 + p x 4 3 ) a − 1 ( 2 3 n x − 1 3 + 4 3 p x − 7 3 ) . {\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 = 1 x 3 − a 2 {\displaystyle y={\dfrac {1}{\sqrt {x^{3}-a^{2}}}}} .
Let u = x 3 − a 2 {\displaystyle u=x^{3}-a^{2}} .
d u d x = 3 x 2 ; y = u − 1 2 ; d y d u = − 1 2 ( x 3 − a 2 ) − 3 2 {\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}}}} . d y d x = d y d u × d u d x = − 3 x 2 2 ( x 3 − a 2 ) 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 − x 1 + x {\displaystyle y={\sqrt {\dfrac {1-x}{1+x}}}} .
Write this as y = ( 1 − x ) 1 2 ( 1 + x ) 1 2 {\displaystyle y={\dfrac {(1-x)^{\frac {1}{2}}}{(1+x)^{\frac {1}{2}}}}} .
d y d x = ( 1 + x ) 1 2 d ( 1 − x ) 1 2 d x − ( 1 − x ) 1 2 d ( 1 + x ) 1 2 d x 1 + 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 ) 1 2 ( 1 + x ) − 1 2 {\displaystyle y=(1-x)^{\frac {1}{2}}(1+x)^{-{\frac {1}{2}}}} and differentiate as a product.)