For the polar curve r = sec(θ)tan(θ), find dy/dx in terms of x.
Correct: A
First, convert the polar equation to rectangular form to simplify finding dy/dx.
Given r = sec(θ)tan(θ).
We know sec(θ) = 1/cos(θ) and tan(θ) = sin(θ)/cos(θ).
So, r = (1/cos(θ)) * (sin(θ)/cos(θ)) = sin(θ)/cos²(θ).
Now, express x and y in terms of θ:
x = rcos(θ) = (sin(θ)/cos²(θ)) * cos(θ) = sin(θ)/cos(θ) = tan(θ).
y = rsin(θ) = (sin(θ)/cos²(θ)) * sin(θ) = sin²(θ)/cos²(θ) = tan²(θ).
From these conversions, we have x = tan(θ) and y = tan²(θ).
Substitute x into the equation for y:
y = x².
Now, find dy/dx:
dy/dx = d/dx (x²) = 2x.
The final answer is A.