Question
Fill in the blanks:
Derivative of $x^2$ w.r.t. $x^3$ is _________.
Derivative of $x^2$ w.r.t. $x^3$ is _________.
✓
Answer
Derivative of $x^2$ w.r.t. $x^3$ is $=\frac{2}{3\text{x}}.$
Solution:
Let $u = x^2$ and $v = x^3$
$\Rightarrow\ \frac{\text{du}}{\text{dx}}=2\text{x}$ and $\frac{\text{dy}}{\text{dx}}=3\text{x}^2$
$\therefore\ \frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}$
$=\frac{2\text{x}}{3\text{x}^2}$
$=\frac{2}{3\text{x}}$
Solution:
Let $u = x^2$ and $v = x^3$
$\Rightarrow\ \frac{\text{du}}{\text{dx}}=2\text{x}$ and $\frac{\text{dy}}{\text{dx}}=3\text{x}^2$
$\therefore\ \frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}$
$=\frac{2\text{x}}{3\text{x}^2}$
$=\frac{2}{3\text{x}}$
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