. Use the differention rule for powers of x to differentiate the following polynomial. f(x) =3xx3+x4 . Use the differentiation rules for exp, ln, sin and cos to differentiate the following function. f(x) = 21n(.r) + 8 cos(x) . Use the differnentiation rule for powers of x to differentiate the following function. f(x) = 4x\"4+ i . Differentiate the following function using the Product rule f(x) = (x2 + 2)e-' . Differentiate the following function using the Quotient rule _ sin(x) gm ex . Differentiate the following function y = ln(3x) . Differentiate the following function using the Chain rule: x2+3x+2 y=e . Differentiate the following function: xcusLt] y=e 1 l. The position of an object subject to uniform circular motion around the origin is given by (see Vector II worksheet, Week 4): 1"(1') = R (cos(mr))l sin(wr) (a) Calculate the velocity and acceleration vectors for this trajectory. (b) Show that (1(1) is perpendicular to v(r) for all r. (c) Show that 11(1) - r0) = w2R2 for all :. Him: Recall that 9112(9) + cos2(6) = l. 9. For the following graphical representation of a function x): (1) Mark the points on the graph where the derivative is zero; (2) Show the regions where the function is increasing (positive derivative) and where it is decreasing (negative derivative); (3) Sketch the rst derivative f'(x); (4) Repeat steps (1H3) on the derivative f'(x) itself to sketch the second derivative f\"(x). 10. Calculate the velocity v(r) and acceleration a(r) of the following vector trajectory r(r):( esintr) ) sin(i'2 + r]