3. Techniques of Differentiation

Derivatives Find and simplify the derivative of the following functions.

g(w) = √w+w / √w-w

Derivatives Find and simplify the derivative of the following functions.

h(w) = w⁵/³ / w⁵/³+1

Derivatives Find and simplify the derivative of the following functions.

g(x) = x⁴/³-1 / x⁴/³+1

Population growth Consider the following population functions.

a. Find the instantaneous growth rate of the population, for t≥0.

p(t) = 600 (t²+3/t²+9)

Population growth Consider the following population functions.

b. What is the instantaneous growth rate at t=5?

p(t) = 600 (t²+3/t²+9)

Population growth Consider the following population functions.

c. Estimate the time when the instantaneous growth rate is greatest.

p(t) = 600 (t²+3/t²+9)

Population growth Consider the following population functions.

e.Use a graphing utility to graph the population and its growth rate.

p(t) = 600 (t²+3/t²+9)

Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).

f(x) = 1/x

Given that f(1) = 5, f′(1) = 4, g(1) = 2, and g′(1) = 3 , find d/dx (f(x)g(x))∣ ∣x=1 and d/dx (f(x) / g(x)) ∣ x=1.

Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

Derivatives Find and simplify the derivative of the following functions.

f(x) = 2e^x-1 / 2e^x+1

Derivatives Find and simplify the derivative of the following functions.

s(t) = t⁴/³ / e^t

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

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

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)