Reference no: EM131076298

Derivatives and implicit differentiation-

(1) Find the derivative of the following functions.

(a) f(x) = x^cosx, defined on the domain (0, ∞).

(b) f(x) = log₅(3x² - 2), defined on the domain (-∞, -√2/3) ∪ (√2/3, ∞)

(c) f(x) = (√x)^x, defined on the domain (0, ∞).

(2) Let f be the function f(x) = ln(x - 1), defined on the domain (1, ∞). Find dⁿf/dxⁿ for any positive integer n.

(3) Find dy/dx by implicit differentiation.

(a) x³ + y³ = 6xy

(b) x sin y + y sin x = 1

(4) Find all points on the curve x²y² + xy = 2 where the slope of the tangent line is -1.

(5) Draw a graph of the curve x² - y² = 1. Use implicit differentiation to find the tangent line at all points except (-1, 0) and (1, 0). What goes wrong for these points? What is the tangent line to the curve at the point (1, 0)?

(6) Draw a graph of the curve x² - y² = 0. Argue that there isn't a good way to define the "tangent line" to the curve at the point (0, 0).

(7) Draw a graph of the curve y² - x⁴ = 0. Use implicit differentiation to find the tangent line at all points except (0, 0). What goes wrong for (0, 0)? Find the tangent line to the curve at (0, 0).