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72

Calculus Made Easy

Let $u=(a^{2}-x^{2})^{-{\frac {1}{6}}}$ and $v=(a^{2}-x^{2})$ .

$u$	$=v^{-{\frac {1}{6}}};\quad {\frac {du}{dv}}=-{\frac {1}{6}}v^{-{\frac {7}{6}}};\quad {\frac {dv}{dx}}=-2x$ .
${\frac {du}{dx}}$	$={\frac {du}{dv}}\times {\frac {dv}{dx}}={\frac {1}{3}}x(a^{2}-x^{2})^{-{\frac {7}{6}}}$ .

Let $w=(a^{2}-x^{2})^{\frac {1}{6}}$ and $z=(a^{2}+x^{2})$ .

$w$	$=z^{\frac {1}{6}};\quad {\frac {dw}{dz}}={\frac {1}{6}}z^{-{\frac {5}{6}}};\quad {\frac {dz}{dx}}=2x$ .
${\frac {dw}{dx}}$	$={\frac {dw}{dz}}\times {\frac {dz}{dx}}={\frac {1}{3}}x(a^{2}+x^{2})^{-{\frac {5}{6}}}$ .

Hence

	${\frac {dy}{dx}}$	$=(a^{2}+x^{2})^{\frac {1}{6}}{\frac {x}{3(a^{2}-x^{2})^{\frac {7}{6}}}}+{\frac {x}{3(a^{2}-x^{2})^{\frac {1}{6}}(a^{2}+x^{2})^{\frac {5}{6}}}}$ ;
or	${\frac {dy}{dx}}$	$={\frac {x}{3}}\left[{\sqrt[{6}]{\frac {a^{2}+x^{2}}{(a^{2}-x^{2})^{7}}}}+{\frac {1}{\sqrt[{6}]{(a^{2}-x^{2})(a^{2}+x^{2})^{5}]}}}\right]$ .

(9) Differentiate $y^{n}$ with respect to $y^{5}$ .

${\frac {d(y^{n})}{d(y^{5})}}={\frac {ny^{n-1}}{5y^{5-1}}}={\frac {n}{5}}y^{n-5}$ .

(10) Find the first and second differential coefficients of $y={\dfrac {x}{b}}{\sqrt {(a-x)x}}$ .

${\frac {dy}{dx}}={\frac {x}{b}}\,{\frac {d{\bigl \{}{\bigl [}(a-x)x{\bigr ]}^{\frac {1}{2}}{\bigr \}}}{dx}}+{\frac {\sqrt {(a-x)x}}{b}}$ .

Let ${\bigl [}(a-x)x{\bigr ]}^{\frac {1}{2}}=u$ and let $(a-x)x=w$ ; then $u=w^{\frac {1}{2}}$ .

${\frac {du}{dw}}={\frac {1}{2}}w^{-{\frac {1}{2}}}={\frac {1}{2w^{\frac {1}{2}}}}={\frac {1}{2{\sqrt {(a-x)x}}}}$ .