Answer
$\frac{5}{3}$ units/s
Work Step by Step
Equation of moving point $ P(x,y)=P(x,\sqrt {x^3+17 })$ is:
$y=\sqrt {x^3+17}$
Taking derivative with respect to time
$\frac{dy}{dt}=\frac{ d( \sqrt {x^3+17}) }{dt}$
$\frac{dy}{dt}=\frac{ d( \sqrt {x^3+17}) }{dx} \frac{dx}{dt}$
$\frac{dy}{dt}=\frac{1}{2}(x^3+17)^{\frac{1}{2}-1} \frac{d(x^3+17)}{dx}\frac{dx}{dt}$
$\frac{dy}{dt}=\frac{1}{2}(x^3+17)^{ -\frac{1}{2}} (3x^2)\frac{dx}{dt}$
$\frac{dy}{dt}=\frac{3}{2} \frac{x^2}{\sqrt {x^3+17}} \frac{dx}{dt}$................... eq(1)
Put $ \frac{dy}{dt}= 2 $ units/s, $x=2$ in equation (1)
$ 2=\frac{3}{2} \frac{2^2}{\sqrt {2^3+17}} \frac{dx}{dt} = \frac{6}{5} \frac{dx}{dt}$
$ 2= \frac{6}{5} \frac{dx}{dt}$
$\frac{dx}{dt}=\frac{5}{3}$ units/s
