Calculus word problem help?

Question

To supply blood to all parts of the body, the human artery system must branch repeatedly. Suppose an artery of radius r branches off from an artery of radius R (r>r) at an angle θ. The energy lost due to friction is approximately this:

E(θ)=(csc θ/r^4)+((1-cot θ)/R^4).

Find the value of θ that minimizes the energy loss.

I'm not asking for the answer, I just need help in finding the derivative for the equation, so that I can solve for θ. That is how the book has the problem stated.

Answer 1

E´(t) =1/r^4*1/sin^2 *(-cost)+1/R^4( 1/tg^2 t)(1+tg^2 t)
As in this equation sin t only appears squared you can put
sin^2t = 1-cos^2t and get an equation in cos t

Answer 2

enable x = distance of base of ladder from domicile. enable y = height of precise of ladder. x² + y² = 29² = 841 while base of ladder is 21 ft from domicile: y² = 841 - 21² y² = 4 hundred y = 20 x² + y² = 841 Differentiate the two components with admire to t 2x dx/dt + 2y dy/dt = 0 y dy/dt = -x dx/dt while base is 21 ft from the domicile, x = 21, y = 20 Base of ladder is pulled removed from the domicile at a cost of four ft/sec: dx/dt = 4 y dy/dt = -x dx/dt 20 dy/dt = -21 * 4 dy/dt = -80 4/20 dy/dt = -4.2 So precise of ladder is shifting down the wall at a cost of four.2 ft/sec -------------------- observe: dy/dt = -4.2 The destructive sign shows that precise of ladder is shifting in downward direction. So we are saying that ladder is shifting at a cost of -4.2 ft/sec OR ladder is shifting DOWN at a cost of four.2/sec