Question

Item 5
A right triangle has base x meters and height h meters, where h is constant and x changes with respect to time t, measured in seconds. The angle θ, measured in radians, is defined by tanθ=hx. Which of the following best describes the relationship between dθdt, the rate of change of θ with respect to time, and dxdt, the rate of change of x with respect to time?

Answer 1

fix your typo, and then use the chain rule
tanθ = h/x
sec^2θ dθ/dt = -h/x^2 dx/dt

if h were also changing, then the quotient rule would also come into play:
sec^2θ dθ/dt = (x dh/dt - h dx/dt)/x^2
Note that since h is constant in your problem, dh/dt = 0

Answer 2

I can not see what you pasted of course
tanθ=h/x, I suspect not h x
h is constant
d/dt tan θ = sec^2 θ dθ/dt = -h/x^2 dx/dt
so
dθ/dt = -(1/sec^2θ) h/x^2 dx /dt
but sec = 1/cos
so
dθ/dt = -cos^2 θ * (h/x^2) dx/dt
and cos θ = x / (h^2+x^2)
etc