(8) Differentiate ${\displaystyle y^{5}}$ with respect to ${\displaystyle y^{2}}$.

(9) Differentiate ${\displaystyle y={\dfrac {\sqrt {1-\theta ^{2}}}{1-\theta }}}$.

The process can be extended to three or more differential coefficients, so that ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {dy}{dz}}\times {\dfrac {dz}{dv}}\times {\dfrac {dv}{dx}}}$

Examples.

(1) If ${\displaystyle z=3x^{4}}$; ${\displaystyle v={\dfrac {7}{z^{2}}}}$; ${\displaystyle y={\sqrt {1+v}}}$, find ${\displaystyle {\dfrac {dv}{dx}}}$.

We have

${\displaystyle {\frac {dy}{dv}}}$	${\displaystyle ={\frac {1}{2{\sqrt {1+v}}}};\quad {\frac {dv}{dz}}=-{\frac {14}{z^{3}}};\quad {\frac {dz}{dx}}=12x^{3}}$.
${\displaystyle {\frac {dy}{dx}}}$	${\displaystyle =-{\frac {168x^{3}}{(2{\sqrt {1+v}})z^{3}}}=-{\frac {28}{3x^{5}{\sqrt {9x^{8}+7}}}}}$.

(2) If ${\displaystyle t={\dfrac {1}{5{\sqrt {\theta }}}}}$; ${\displaystyle x=t^{3}+{\dfrac {t}{2}}}$; ${\displaystyle v={\dfrac {7x^{2}}{\sqrt[{3}]{x-1}}}}$, find ${\displaystyle {\dfrac {d\phi }{dx}}}$.

an expression in which ${\displaystyle x}$ must be replaced by its value, and ${\displaystyle t}$ by its value in terms of ${\displaystyle \theta }$.

(3) If ${\displaystyle \theta ={\dfrac {3a^{2}x}{\sqrt {x^{3}}}}}$; ${\displaystyle \omega ={\dfrac {\sqrt {1-\theta ^{2}}}{1+\theta }}}$; and ${\displaystyle \phi ={\sqrt {3}}-{\dfrac {1}{\omega {\sqrt {2}}}}}$, find ${\displaystyle {\dfrac {d\phi }{dx}}}$.