Let u = ( a 2 − x 2 ) − 1 6 {\displaystyle u=(a^{2}-x^{2})^{-{\frac {1}{6}}}} and v = ( a 2 − x 2 ) {\displaystyle v=(a^{2}-x^{2})} .
Let w = ( a 2 − x 2 ) 1 6 {\displaystyle w=(a^{2}-x^{2})^{\frac {1}{6}}} and z = ( a 2 + x 2 ) {\displaystyle z=(a^{2}+x^{2})} .
Hence
(9) Differentiate y n {\displaystyle y^{n}} with respect to y 5 {\displaystyle y^{5}} .
d ( y n ) d ( y 5 ) = n y n − 1 5 y 5 − 1 = n 5 y n − 5 {\displaystyle {\frac {d(y^{n})}{d(y^{5})}}={\frac {ny^{n-1}}{5y^{5-1}}}={\frac {n}{5}}y^{n-5}} .
(10) Find the first and second differential coefficients of y = x b ( a − x ) x {\displaystyle y={\dfrac {x}{b}}{\sqrt {(a-x)x}}} .
d y d x = x b d { [ ( a − x ) x ] 1 2 } d x + ( a − x ) x b {\displaystyle {\frac {dy}{dx}}={\frac {x}{b}}\,{\frac {d{\bigl \{}{\bigl [}(a-x)x{\bigr ]}^{\frac {1}{2}}{\bigr \}}}{dx}}+{\frac {\sqrt {(a-x)x}}{b}}} .
Let [ ( a − x ) x ] 1 2 = u {\displaystyle {\bigl [}(a-x)x{\bigr ]}^{\frac {1}{2}}=u} and let ( a − x ) x = w {\displaystyle (a-x)x=w} ; then u = w 1 2 {\displaystyle u=w^{\frac {1}{2}}} .
d u d w = 1 2 w − 1 2 = 1 2 w 1 2 = 1 2 ( a − x ) x {\displaystyle {\frac {du}{dw}}={\frac {1}{2}}w^{-{\frac {1}{2}}}={\frac {1}{2w^{\frac {1}{2}}}}={\frac {1}{2{\sqrt {(a-x)x}}}}} .
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