do you see where you went wrong? you forgot to put back the sec(5x) when you differentiated. and the sec(5x)tan(5x) is not the part that should be raised to the power

I'm having trouble getting the following word question:

A police officer in a patrol car is approaching an intersection at 25m/s. When he is 210 m from the intersection, a truck crosses the intersection travelling at right angles to the police car's path at a rate of 25m/s. If the oficer focusses his spotlight on the truck, how fast is the light beam turning 3 seconds later assuming that both vehicles continue at their same rates.

I would appreciate if someone could take a quick look at this question, maybe just give me a hint of what kind of approach to take, I have a feeling I can figure it out once I have a start.

The following two I thought I had figured out, but then I checked my answer with a graphing calculator and it didn't make any sense.

Find dy/dx of 1-2cos^2(x) = 4sin(x)cos(x)+y^2
I simplified this one to sin^2(x) = sin(4x)+y^2
and then tried to differentiate from there but it didn't work out, and I tried to differentiate the original as well. Was the original simplification incorrect?

Find dy/dx of (1/(1+cos(x)))
I tried this out and had something that looked good, but then I checked with a calculator and the derivative didn't mirror the slope of the equation at all.

Thanks for looking. Hopefully I can get all this stuff figured out soon and answer some questions for other people.

I'm having trouble getting the following word question:

A police officer in a patrol car is approaching an intersection at 25m/s. When he is 210 m from the intersection, a truck crosses the intersection travelling at right angles to the police car's path at a rate of 25m/s. If the oficer focusses his spotlight on the truck, how fast is the light beam turning 3 seconds later assuming that both vehicles continue at their same rates.

I would appreciate if someone could take a quick look at this question, maybe just give me a hint of what kind of approach to take, I have a feeling I can figure it out once I have a start.

you asked for a hint, so that's what i'll give. see the diagram below, this is a related rates problem.

Find dy/dx of 1-2cos^2(x) = 4sin(x)cos(x)+y^2
I simplified this one to sin^2(x) = sin(4x)+y^2
and then tried to differentiate from there but it didn't work out, and I tried to differentiate the original as well. Was the original simplification incorrect?

I'm working on that word problem currently. I would appreciate if someone could take a quick look at my solutions to the following problems. I have a feeling I messed something up, because I'm good at messing things up.

dy/dx of y=[csc^2(x)]-[3x^2][cot^2(x)]
dy/dx=2(cscx)d/dx(cscx) - 6x(2(cotx))d/dx(cotx) dy/dx=2(cscx)(-cscxcotx) - 12x(cotx)(-csc^2(x))

dy/dx of y = csc((x^2-2)^3)
dy/dx= [-csc(x^2-2)^3][cot(x^2-2)^3][d/dx(x^2-2)^3]
dy/dx= [-csc(x^2-2)^3][cot(x^2-2)^3][3(x^2-2)^2][d/dx(x^2)] dy/dx= [-csc(x^2-2)^3][cot(x^2-2)^3][3(x^2-2)^2][2x]