'I. Chains Inc. is in the business of making and selling chains. Let c(t)be the number of miles of chain produced after t 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 q'(t) How does q'(t) relates to p'(c)and c'(t)? 2. Use Desmos to graph the function y3 + ymz + 332 _ 33,2 = 0 and estimate the slope of the tangent line at (1,1). Then find it! using implicit differentiation and plug in :1; = 1 and y = 1. Compare and discuss the estimated slope with the slope you found analytically. 3. Let f(g;) = (33:2 + 1)). Find f'(;z:) in 3 different ways by following the instructions below in parts a, b and c: a) Develop the identity f(:r:) then take the derivative. b) View f(:r:) as (3332 + 1)(3g;2 + 1) and use the product rule to find f'(1:). c) Apply the chain rule directly to the expression f(;1:) = (3:32 + 1)2. d) Are your answers in parts a, b, c the same? Why or why not? 4. Find gfor 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 + 32y _ :1: = 65 6. Find $1 for the equation 1:312 32y = 2 7. Find f' (a) for the function f (x) = V-3x4 - 2 8. Find f (a) for the function f (x) = (5x2 + 3)4 9. Find f (x) for the function f(x) = sin?(cos(4x)) 10. Find f (x ) for the following functions: a) f(x) = cos(In4x3)) b) f (x) = e(423+5)2