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I can't seem to get the correct answer for this problem:

A particle is moving along the curve $y=4\sqrt{4x+5}.$ As the particle passes through the point $(5,20),$ its $x$-coordinate increases at a rate of $2$ units per second. Find the rate of change of the distance from the particle to the origin at this instant.

Here's my attempt at the problem (I'm not sure where I went wrong):

enter image description here

Thanks for your help!

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Your formula for $\frac{dy}{dt}$ is incorrect. You forgot the apply one instance of the chain rule, to take the derivative of $4x+5$, so you are missing a factor of $4$. Or something like that, since you apparently corrected the $2$ to a $4$, but it should have been $8$. It should be

$$\begin{align} \frac{dy}{dt} &= \frac d{dt}\left(4\sqrt{4x+5}\right) \\[3 ex] &= 4\frac{1}{2\sqrt{4x+5}}\cdot 4\frac{dx}{dt} \\[2 ex] &= \frac{8}{\sqrt{4x+5}}\frac{dx}{dt} \end{align}$$

You can continue from there.

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