(Derivatives, 16pt) Calculate the derivatives of the following functions from the definition of the derivative (not using any other laws or rules).

5. (Derivatives, 16pt) Calculate the derivatives of the following functions from the definition of the derivative (not using any other laws or rules). Show the details of your work. You can check your work in Matlab using the diff() function (you need syms for that; e.g., try syms x, f(x) = x 2, diff(f) ). a) [5pt] Find the derivative of 1/x for every x. Hint: work from the definition; use common denominator to evaluate difference of two fractions and simplify expression. b) [5pt] Find the derivative of abs(x) for every x. Hint: You'll have to distinguish three cases depending on the value of x (what are they?). Work with the definition of the derivative as the limit of (f(xth)-f(x))/h as h goes to 0, where f (x) = abs(x). c) [6pt] Find the derivative of f(x) = sin(x) for every x. Hint: work with the definition of the derivative and plug in f(x) = sin(x). To deal with the term sin(xth) you'll get, work with the angle sum identity for sine). Then separate the x and h parts (as we did for ex). If you need to compute any specific limits, use Matlab for that (again, similar to what we did for ex)