These are the collection of some questions which i want to learn. Please answer in detail(step by step) and also include diagram or graph if needed. Please provide typed work as many times handwriting makes hard to understand.

anrite and explain an example ofa function whose derivative can be found by u5ing the following rules a} Product rule and Special function differentiation rulea o} Power rule. quotient rule. and chain rule c} Chain rule twice d} Implicit differentiation and special function differentiation rule I 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 19(5) be the profit as a function ofthe number of miles of chain produced and let gift) be the prot as a function ofthe number of hours of production. Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret {use complete sentences} each of the following (include units}, gift). FTC)- and (fa). How does q') relates to p'(c)and (Hf)? - 2 2 2 _ - - _ EH - in Use Desmos to graph the function y3 + ya, + 5,3 _ 3y _ f] and estimate the slope ofthetangent line at{ 1.1}. Then nd d3: usmg implicit differentiation and plug in a: = 1 and y = 1. Compare and discuss the estimated slope with the slope you found analytically. a Let ffgg) = (33:2 + 1)? Find f'[$) in 3 different ways by following the instructions below in parts a. b and c: a] Develop the identity fICI) then take the derivative. b} View ffl') as (355-2 + \"(33:2 -|- 1) and use the product rule to nd f'($). (1 Apply the chain rule directly to the expression f(;1:) = (3352 + 1)2. :21} Are your answers in parts a, b. c the same? Why or why not? 3 Find g: for the equation 33:2 _ cosy = 3:3 , Find the equation of the tangent line that passes through point {'I .2} to the graph 8313 + 3:23..- a: = 65 - d - 2 2 _ o FIndEEfor the equationmy :17 y _ 2 V\" 3a:4 2 g Find flft) forthe functionf(_r,g) .' Find f'(:1:) forthe functionffgc) = (5332 + 3)4 Find f' (a ) for the function f(x) = sin(cos(4x)) Find f' (a ) for the following functions: a) f(@) = cos(In4r b) f(x) = (12 +5)