1, Chains Inc. is in the business of making and selling chains, Let c(t)be the number of miles of chain produced after 1 hours of production. Let p(c) be the profit as a function of the number of miles of chain produced and let q(t) be the profit as a function of the number of hours of production Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret (use complete sentences) each of the following (include units), c'(t), p'(c), and ({0} How does q'(t) relates to p'(c)and c'(t)? 2, Use Desmos to graph the function y3 + yxz + 132 _ 33/2 = 0 and estimate the slope of the tangent line at (4,1), Then find g:- using implicit differentiation and plug in g; = _1 and y = 1. Compare and discuss the estimated slope with the slope you found analytically. 3, Let f(;1;) = (3132 + 1)2. Find fl(.'t) in 3 different ways by following the instructions below in parts a, b and c: a) Develop the identity f(1') then take the derivative. b) View f(z) as (fix2 + 1x332 + 1) and use the product rule to find f'(.1:) c) Apply the chain rule directly to the expression f(3:) = (33:2 + 1)2. d) Are your answers in parts a, b, c the same.7 Why or why not? 4, Find g for the equation 3y2 _ cosy = 3:3 5, Find the equation of the tangent line that passes through point (1,2) to the graph 83,3 + x2y _ 1; = 65 6. Find a: for the equation my? $21,; = 2 7. Find f'(1) for the function ftp) = m 8, Find f'(1) for the function f($) = (51:2 + 3)4 9, Find f'(:t) for the function ftp) = sin2(cos(4:v)) 10. Find f'(a:) for the following functions: a)f(a:) = cos(ln4z3) b) f(2) = ear-3+5)