If anyone could help me with these questions, please do so! Also please provide detailed solutions and write down solutions on paper. Thank you!

1. Find the derivative for each of the following functions. a. f (a) = 27 b. f (x) = 2x3 c. f (x) = -220 d. f (ac) = e. f (20) = 2-3/5 f. f (20) = Vac g. f (2) = V23 n. f (ac) = Va2 i. f (2c) Vi 2. Choose any one(1) of the functions above and find the derivative of that functions using the method of "first principles". Please pick any function that you like.1. Launch Desmos and Microsoft Excel. 2. Using both applications (similar to work that you completed in Unit 4), graph the function f(x) = 8x3 - 5x3+ 7x. Determine the slope of the tangent line and various points along the x-axis. Place the values in the table below. Repeat the above steps for the functions. Place the values in the table below. x - value Slope of f(x) = 8x - 5x2+ 7x g(x) = 3x3- 1 h(t) = 4t + 5t + 2 y = 7x - 2x + 3 -3 -2 WNOM 7. Enter the values of x and the corresponding slope values for each of the functions into Microsoft Excel. Perform a regression to find the curve of best fit. Use your knowledge to hypothesize the "type" of equation that would best approximate this new function for the derivative (je. use the Power Rule).8. Complete the table to compare the original function with its derivative. Function f(x) = 8x3 - 5x- + 7x g(x) = 3x3- 1 h(t) = 4+ + 5tz+ 2 y = 7x - 2x + 3 Regression Equation f' (x) = g (x)= h'(t) = (Derivative) 9. What do you notice about the functions and the equations that represent their derivatives? Formulate a hypothesis of the sum and difference rule for derivatives.1. Determine the derivative of the following functions. You can use the derivative rules from this section and not find these derivatives by first principles. Simplify first, if necessary, and then find the derivative. a. f (x) = 3x2 - 5x5 + 202 + 7 b. g (2) = 2Vx - Va+ Va c. p(a) = 3 - 2ex - -sin (2x) + 3 cos x d. m (x) = (3x +2)2 e. s (t) = 4+3 + 5+4 1 2 f. y = 3x2 (4x5 - 2x3) 2. Using the definition of derivative, prove the constant multiplier rule. That is, prove the derivative of g(x) = c f(x) is g'(x) = c f'(x). Follow the same format that was used in the content section to prove the derivative of a constant rule. 3. Determine the equation of the tangent line to the graph of f(x) = -x3 + 3x2 - 2 at x = 1. Confirm the this equation by graphing (DESMOS) both the function, the common point and tangent line. Include a copy of the graphs in your solution. 4. Suggest a function that would have a derivative of f'(x) = 12x2 + 6x - 5. Explain how you arrived at your