Please explain all steps in detail from YOUR OWN WORK for good feedback.

4. 5. 6. The natural logarithmic function In x is used in many differentiation and integration problems that do not at first glance appear to involve logarithms. a) Differentiate In x. (Use the fact that elnx = x.) b) Differentiate Inlxl. (Consider the two separate cases when x > 0 and when x < O.) Prove the Power Rule for any real number n. Show that if n is any real number and f(x) then f' (x) = nxn-I. (Express xn as a power of e.) We have learned how to differentiate all the basic types of functions, but a function such as e2x(2x 1)' (x3 + 7x) = xn, requires four chain rules inside vo product rules, and a giant quotient rule! Instead, use logarithmic differentiation to express dy /dx. (Take the natural logarithm of both sides and then differentiate both sides with respect to x.)