3. Techniques of Differentiation

Derivatives Find and simplify the derivative of the following functions.

g(w) = √w+w / √w-w

Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

f(x) = x(x-1)

Assume that functions f and g are differentiable with f(2) = 3, f'(2) = −1, g(2) = −4, and g'(2) = 1. Find an equation of the line perpendicular to the line tangent to the graph of F(x) = (f(x) + 3) / (x − g(x)) at x = 2.

Assume that functions f and g are differentiable with f(1) = 2, f'(1) = −3, g(1) = 4, and g'(1) = −2. Find the equation of the line tangent to the graph of F(x) = f(x)g(x) at x = 1.

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

d/dx (f(x)g(x)) | x=4

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

75. y = x³ log₁₀ x

Generalizing the Product Rule The Derivative Product Rule gives the formula

d/dx (uv) = u (dv/dx) + (du/dx) v

for the derivative of the product uv of two differentiable functions of x.

b. What is the formula for the derivative of the product u₁u₂u₃u₄ of four differentiable functions of x?

9–61. Evaluate and simplify y'.

y = e^6x sin x

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

b. y = f(x) / g(x)

15–48. Derivatives Find the derivative of the following functions.

y = e^x x^e

Find the derivatives of the functions in Exercises 1–42.

__

s = √ t .

1 + √ t

Find and simplify the derivative of the following functions.

y = (3t−1)(2t−2)^-1

Find and simplify the derivative of the following functions.

h(x) = (x − 1)(x³+ x² + x+1)

Derivatives Find and simplify the derivative of the following functions.

h(w) = w⁵/³ / w⁵/³+1