22 Which of the following rules holds true for any continuous and differentiable functions a(r), b(x), c(x), and d(x) ? (Use the product rule and quotient rule to verify.) Select one alternative: d a a'bed - ab'c'd' O dx Abcd b2 c2 d2 O a a'bed + ab'cd + abc'd + abcd' = bcd b2 c2 d2 O d a a'bed - ab'cd - abc'd - abcd' = bed 62 c2 d2 d a a'bed + ab'c'd' O da b2 c2 d221 Which of the following limits corresponds to the derivative of f(z) = 4e-2:? Select one alternative: 4e-2x-2h - 4e-2x O lim h-0 h 40-22th - 4p-21 O lim h-+0 h 8e-27th + 8e-2x O lim h-+0 h 8e-2x-2h + 8e-2x O lim h-+0 h24 Consider the implicit relation between c and y defined by f(x) + f(y) + f(xy) = 0 for some continuous and differentiable function f(@). Which of the following expressions gives the correct derivative? Select one alternative: O dy f'(x) + yf'(xy) dx f'(y) + xf'(xy) f'(y) + xf'(xy) O dy dx f' (x) + yf'(xy) dy f'(y) + xf'(xy) O da f'(x) +yf'(xy) dy f'(x) + yf'(xy) O f'(y) + xf(ry)25 Consider the relation x2 = y" + 1. In which of the 4 quadrants is dy > 0? You might find the following diagram useful to remind you which quadrant is which. 2 1 X 3 4 Select one alternative: O 1 & 2 04 & 1 O2 & 3 03 & 423 Which of the following expressions is the correct derivative of the expression 5e"(2) + 3/u(x)| for some continuous and differentiable function u(x) when u(x) * 0? In the following answers sign(u(c) ) just extracts the sign of the function u(). It could also be represented as follows: -1, if u(x) 0 Select one alternative: O 5u'(x)eu(2) + 3u'(x) sign(u(x)) 0 5u'(2)e"(z) + 3/u'(2)| 0 5e" (2) + 3/u'(2)| ( 5e"(z) + 3u'(x) sign(u(x))