Please answer all questions and show all work. Answer key is shown in the pictures, I need help showing the work, so please show all work and provide a step by step breakdown / explanation and write neatly as possible. Thank you!

\f4. Find the derivative of each function. Simplify the function before you take the derivative a) v(t) = 5+3 - # b) f (x) = 15 V x - 37 -7 c) f(x) = 3x-4x-5 Vx d) g(t) = 9Vth_8 2 4 + V2 e) h(t) = 5t*(7012) a) v' (t) = 15t2 + 2; b) f' (x) = 5x-2/3+ 2x-4/3; c) f'(x)=2Vx-2+5 e) h'(t) = +3t2 Vx 2Vx3 " d) g' (t) = 6t /3 + 32t-5 ; 5. Use the product, quotient and chain rules as needed to find each derivative a) d 2 x 6 + 10 b) - d x dx c) 5x3 7x +5 d)- d dx 2x+1 dx - (x2 -4)(5x-1)4 e) . f) d (10t2+5t)2 at 5 8) at 2+ + 1, a) 4x (2x +10) b) c)15x2 V7x +5+3 35x3 (2x+ 1) 2 > 2V7x+5 " d) 20(x2 -4)(5x -1)3 +2x(5x-1)4 6 (2t + 1) 2 6. Find the equation of the tangent line to the curve at the given point. Write the answer in slope-intercept form a) f(x) = x4 - x2 at the point (1, 0) b) g(x) = z at x = 2 c)y = x - 2vx at x =4 a) y = 2x -2 b) y = 1/2x-3/2 cy = 1 2 x - 2 7. Find the Critical Points (points where the tangent line is horizontal) of each given function a) f (x ) = x4 - x2 b) g(x) = 2 c)y = x - 2Vx a) (0,0) and (+- 1 1 2' A ) b) tangent line is never horizontal c) (1,-1)1 1. A company that manufactures ski jackets estimates that their xed costs are $21 ,900 and variable costs of $24 per jacket. A survey indicates that a price of $140 results in a demand of demand of 300 jackets and that a price of $100 will result in a demand of 500 jackets. a) Find the price-demand equation assuming a linear relationship b) Find the revenue function and determine the domain its domain using the price-demand equation c) Find the total cost function d) Find the prot function and the marginal prot function e) Evaluate the marginal prot at production levels of 300 and 600 jackets and interpret your result being specic about quantities, units, and increasing/decreasing f) Determine the break-even points Answers: a) p = ix + 200 b) R(x) =x2+200x D: 0 5x5 1000 c) C(x) = 21,900 + 24x d) P(x) = x2 + 176x 21,900 P'(x) = x + 176 e) P'(300) = 5 dollars _ dollars 0 6 /jacket P'(600) _ _64 /jacket 9) Break-even points at production levels of 150 and 730 jackets