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Question:

Differentiate $x^{2}$ with respect to $x^{3}$

Solution:

Let $u=x^{2}$ and $v=x^{3}$

$\Rightarrow \frac{d u}{d x}=2 x$ and $\frac{d v}{d x}=3 x^{2}$

$\therefore \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 x}{3 x^{2}}=\frac{2}{3 x}$

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