5 questions derivatives
1. (2) Find the slope of the line tangent to the curve given by x3 + 2y2 = 3x + 2y−2 at the point P(−2,1).
2. (4) Consider the function y de ned implicitly as a function of x by the given equation:
(a.) xsin(y2) + y sin(3x) = 3; (b.) yex + xey = xy.
Compute
dy dx
by implicit dierentiation for each of the above functions y.
3. (4) Consider the function y de ned implicitly as a function of x by the equation: y2 −4x3y + 6×4 = 18. (a) (2) Compute dy dx by implicit dierentiation. (b) (2) Find
all equations of all lines tangent to the graph of y at the point on the curve where x = 1.
4. (4) Find the derivative of the following functions by logarithmic dierentiation only. Show all your steps and you do not need to simplify your answer. (a) f(x) =
lncos2(x)p1 + x4 (b) f(x) =x2 −81/3√x3 + 1 x6 −7x + 5 .
5. (4) Find the second derivative of the following functions. Simplify as much as you can. (a) f(x) = xex2 (b) g(x) = x3 ln(x) (c) h(x) = x x2 + 2
.
