We get
θ = 3 a 2 x − 1 2 ; ω = 1 − θ 1 + θ ; {\displaystyle \theta =3a^{2}x^{-{\frac {1}{2}}};\quad \omega ={\sqrt {\frac {1-\theta }{1+\theta }}};\quad } and ϕ = 3 − 1 2 ω − 1 {\displaystyle \quad \phi ={\sqrt {3}}-{\frac {1}{\sqrt {2}}}\omega ^{-1}} .
d θ d x = − 3 a 2 2 x 3 ; d ω d θ = − 1 ( 1 + θ ) 1 − θ 2 {\displaystyle {\frac {d\theta }{dx}}=-{\frac {3a^{2}}{2{\sqrt {x^{3}}}}};\quad {\frac {d\omega }{d\theta }}=-{\frac {1}{(1+\theta ){\sqrt {1-\theta ^{2}}}}}}
(see example 5, p. 69); and
d ϕ d ω = 1 2 ω 2 {\displaystyle {\frac {d\phi }{d\omega }}={\frac {1}{{\sqrt {2}}\omega ^{2}}}} .
So that d θ d x = 1 2 × ω 2 × 1 ( 1 + θ ) 1 − θ 2 × 3 a 2 2 x 3 {\displaystyle {\dfrac {d\theta }{dx}}={\dfrac {1}{{\sqrt {2}}\times \omega ^{2}}}\times {\dfrac {1}{(1+\theta ){\sqrt {1-\theta ^{2}}}}}\times {\dfrac {3a^{2}}{2{\sqrt {x^{3}}}}}} Replace now first ω {\displaystyle \omega } , then θ {\displaystyle \theta } by its value.
Exercises VII
You can now successfully try the following. (See page 257 for Answers.)
(1) If u = 1 2 x 3 {\displaystyle u={\frac {1}{2}}x^{3}} ; v = 3 ( u + u 2 ) {\displaystyle v=3(u+u^{2})} ; and w = 1 v 2 {\displaystyle w={\dfrac {1}{v^{2}}}} , find d w d x {\displaystyle {\dfrac {dw}{dx}}} .
(2) If y = 3 x 2 + 2 {\displaystyle y=3x^{2}+{\sqrt {2}}} ; z = 1 + y {\displaystyle z={\sqrt {1+y}}} ; and v = 1 3 + 4 z {\displaystyle v={\dfrac {1}{{\sqrt {3}}+4z}}} , find d v d x {\displaystyle {\dfrac {dv}{dx}}} .
(3) If y = x 3 3 {\displaystyle y={\dfrac {x^{3}}{\sqrt {3}}}} ; z = ( 1 + y ) 2 {\displaystyle z=(1+y)^{2}} ; and u = 1 1 + z {\displaystyle u={\dfrac {1}{\sqrt {1+z}}}} , find d u d x {\displaystyle {\dfrac {du}{dx}}} .