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Chain Rule of Differentiation

If a=x y p-...

Question

If $a = x y^{p - 1}, b = x y^{q - 1}$ and $c = x y^{r - 1},$ prove that $a^{q - r} b^{r - p} c^{p - q} = 1$

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Solution

Given:
$a = x y^{p - 1}, b = x y^{q - 1}$ and $c = x y^{r - 1},$

LHS $= a^{q - r} b^{r - p} c^{p - q}$

$= (x y^{p - 1})^{q - r} (x y^{q - 1})^{r - p} (x y^{r - 1})^{p - q}$
$= x^{q - r + r - p + p - q} y^{q p + r - q - p r + q r + p - q p - r + p r + q - p - q r}$
$= x^{0} y^{0} = 1$

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