CPM Homework Help : CALC Problem 7-212

A particle moves along the curve $y = 2x$ so that its $x$ -coordinate increases at the constant rate of $5$ units per second. How fast is its angle of inclination $θ$ changing when $x = 2$ ?

(As shown in the diagram, $θ$ is the angle between the segment joining the point to the origin and the $x$ -axis.)

Hint:

$\text{tan}\theta =\frac{\text{opposite side}}{\text{adjacent side}}$
What are the opposite and adjacent sides here?

Step:

To obtain $θ^\prime$ , you will have to use implicit differentiation in terms of $t$ , since the $x$ -coordinate and $θ$ are changing over time.

Answer:

$2\text{ln}2-1\approx 0.386$