(7) Differentiate y = ( x + x 2 + x + a ) 3 {\displaystyle y=(x+{\sqrt {x^{2}+x+a}})^{3}} .
Let x + x 2 + x + a = u {\displaystyle x+{\sqrt {x^{2}+x+a}}=u} .
d u d x = 1 + d [ ( x 2 + x + a ) 1 2 ] d x {\displaystyle {\frac {du}{dx}}=1+{\frac {d\left[(x^{2}+x+a)^{\tfrac {1}{2}}\right]}{dx}}} .
y = u 3 {\displaystyle y=u^{3}} ; and d y d u = 3 u 2 = 3 ( x + x 2 + x + a ) 2 {\displaystyle {\frac {dy}{du}}=3u^{2}=3\left(x+{\sqrt {x^{2}+x+a}}\right)^{2}} .
Now let ( x 2 + x + a ) 1 2 = v {\displaystyle (x^{2}+x+a)^{\tfrac {1}{2}}=v} and ( x 2 + x + a ) = w {\displaystyle (x^{2}+x+a)=w} .
(8) Differentiate y = a 2 + x 2 a 2 − x 2 a 2 − x 2 a 2 + x 2 3 {\displaystyle y={\sqrt {\dfrac {a^{2}+x^{2}}{a^{2}-x^{2}}}}{\sqrt[{3}]{\dfrac {a^{2}-x^{2}}{a^{2}+x^{2}}}}} .
We get