3. Techniques of Differentiation

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(t) = 6√t

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

f(s) = √s/4

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(x) = 6x⁵ - 5/2 x² + x + 5

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

s(t) = 4√t - 1/4t⁴+t+1

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

b. How fast will the city be growing when it reaches a size of 38 mi²?

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(x) = (√x+1)(√x-1)

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(w) = w³-w/w

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

y = 12s³-8s²+12s/4s

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

h(x) = √x (√x-x³/²)

If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>

a. (f^-1)'(7)

If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>

b. (f^-1)'(3)

Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>

c. (f^-1)'(f(2))

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(s) = 2√s-1; a=25