(7) y = ( x + 3 ) 2 x − 2 {\displaystyle y=(x+3)^{2}{\sqrt {x-2}}} .
log ϵ y = 2 log ϵ ( x + 3 ) + 1 2 log ϵ ( x − 2 ) . 1 y d y d x = 2 ( x + 3 ) + 1 2 ( x − 2 ) ; d y d x = ( x + 3 ) 2 x − 2 { 2 x + 3 + 1 2 ( x − 2 ) } . {\displaystyle {\begin{aligned}\log _{\epsilon }y&=2\log _{\epsilon }(x+3)+{\tfrac {1}{2}}\log _{\epsilon }(x-2).\\{\frac {1}{y}}\,{\frac {dy}{dx}}&={\frac {2}{(x+3)}}+{\frac {1}{2(x-2)}};\\{\frac {dy}{dx}}&=(x+3)^{2}{\sqrt {x-2}}\left\{{\frac {2}{x+3}}+{\frac {1}{2(x-2)}}\right\}.\end{aligned}}}
(8) y = ( x 2 + 3 ) 3 ( x 3 − 2 ) 2 3 {\displaystyle y=(x^{2}+3)^{3}(x^{3}-2)^{\frac {2}{3}}} .
log ϵ y = 3 log ϵ ( x 2 + 3 ) + 2 3 log ϵ ( x 3 − 2 ) ; 1 y d y d x = 3 2 x ( x 2 + 3 ) + 2 3 3 x 2 x 3 − 2 = 6 x x 2 + 3 + 2 x 2 x 3 − 2 . {\displaystyle {\begin{aligned}\log _{\epsilon }y&=3\log _{\epsilon }(x^{2}+3)+{\tfrac {2}{3}}\log _{\epsilon }(x^{3}-2);\\{\frac {1}{y}}\,{\frac {dy}{dx}}&=3{\frac {2x}{(x^{2}+3)}}+{\frac {2}{3}}{\frac {3x^{2}}{x^{3}-2}}={\frac {6x}{x^{2}+3}}+{\frac {2x^{2}}{x^{3}-2}}.\end{aligned}}}
(For if y = log ϵ ( x 2 + 3 ) {\displaystyle y=\log _{\epsilon }(x^{2}+3)} , let x 2 + 3 = z {\displaystyle x^{2}+3=z} and u = log ϵ z {\displaystyle u=\log _{\epsilon }z} .
d u d z = 1 z ; d z d x = 2 x ; d u d x = 2 x x 2 + 3 . {\displaystyle {\frac {du}{dz}}={\frac {1}{z}};\quad {\frac {dz}{dx}}=2x;\quad {\frac {du}{dx}}={\frac {2x}{x^{2}+3}}.}
Similarly, if v = log ϵ ( x 3 − 2 ) {\displaystyle v=\log _{\epsilon }(x^{3}-2)} , d v d x = 3 x 2 x 3 − 2 {\displaystyle {\dfrac {dv}{dx}}={\dfrac {3x^{2}}{x^{3}-2}}} ) and
d y d x = ( x 2 + 3 ) 3 ( x 3 − 2 ) 2 3 { 6 x x 2 + 3 + 2 x 2 x 3 − 2 } . {\displaystyle {\frac {dy}{dx}}=(x^{2}+3)^{3}(x^{3}-2)^{\frac {2}{3}}\left\{{\frac {6x}{x^{2}+3}}+{\frac {2x^{2}}{x^{3}-2}}\right\}.}
(9) y = x 2 + a 2 x 3 − a 3 {\displaystyle y={\dfrac {\sqrt[{2}]{x^{2}+a}}{\sqrt[{3}]{x^{3}-a}}}} .
log ϵ y = 1 2 log ϵ ( x 2 + a ) − 1 3 log ϵ ( x 3 − a ) . 1 y d y d x = 1 2 2 x x 2 + a − 1 3 3 x 2 x 3 − a = x x 2 + a − x 2 x 3 − a a n d d y d x = x 2 + a 2 x 3 − a 3 { x x 2 + a − x 2 x 3 − a } . {\displaystyle {\begin{aligned}\log _{\epsilon }y&={\frac {1}{2}}\log _{\epsilon }(x^{2}+a)-{\frac {1}{3}}\log _{\epsilon }(x^{3}-a).\\{\frac {1}{y}}\,{\frac {dy}{dx}}&={\frac {1}{2}}\,{\frac {2x}{x^{2}+a}}-{\frac {1}{3}}\,{\frac {3x^{2}}{x^{3}-a}}={\frac {x}{x^{2}+a}}-{\frac {x^{2}}{x^{3}-a}}\\and\quad \quad \quad {\frac {dy}{dx}}&={\frac {\sqrt[{2}]{x^{2}+a}}{\sqrt[{3}]{x^{3}-a}}}\left\{{\frac {x}{x^{2}+a}}-{\frac {x^{2}}{x^{3}-a}}\right\}.\end{aligned}}}
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![{\displaystyle y={\dfrac {\sqrt[{2}]{x^{2}+a}}{\sqrt[{3}]{x^{3}-a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f82a785075b241ac7e1a6b155b105eaf6835a2)
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![{\displaystyle {\begin{aligned}\log _{\epsilon }y&={\frac {1}{2}}\log _{\epsilon }(x^{2}+a)-{\frac {1}{3}}\log _{\epsilon }(x^{3}-a).\\{\frac {1}{y}}\,{\frac {dy}{dx}}&={\frac {1}{2}}\,{\frac {2x}{x^{2}+a}}-{\frac {1}{3}}\,{\frac {3x^{2}}{x^{3}-a}}={\frac {x}{x^{2}+a}}-{\frac {x^{2}}{x^{3}-a}}\\and\quad \quad \quad {\frac {dy}{dx}}&={\frac {\sqrt[{2}]{x^{2}+a}}{\sqrt[{3}]{x^{3}-a}}}\left\{{\frac {x}{x^{2}+a}}-{\frac {x^{2}}{x^{3}-a}}\right\}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90d37656d16be825f00c107917dfde093256e7e3)
