# What is the derivative of sqrt(200-x^3)?

To compute the derivative of the function $f \left(x\right) = \sqrt{200 - {x}^{3}}$ we can apply the chain rule (in both Lagrange's and Leibniz's notations):

$\left[h \left(k \left(x\right)\right)\right] ' = h ' \left(k \left(x\right)\right) \cdot k ' \left(x\right)$

$\frac{d}{\mathrm{dx}} h \left(k \left(x\right)\right) = \frac{d}{\mathrm{dy}} h \left(y\right) {|}_{y = k \left(x\right)} \cdot \frac{d}{\mathrm{dx}} k \left(x\right)$

I'll adopt Lagrange's notation. In our case $h \left(y\right) = \sqrt{y}$ and $k \left(x\right) = 200 - {x}^{3}$, so

$h ' \left(y\right) = \frac{1}{2 \sqrt{y}}$
$k ' \left(x\right) = - 3 {x}^{2}$

By chain rule:

$f ' \left(x\right) = \left[h \left(k \left(x\right)\right)\right] ' = h ' \left(k \left(x\right)\right) \cdot k ' \left(x\right) = \frac{1}{2 \sqrt{200 - {x}^{3}}} \cdot \left(- 3 {x}^{2}\right) = - \frac{3}{2} {x}^{2} / \left(\sqrt{200 - {x}^{3}}\right)$