3. Techniques of Differentiation

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

81. y = log₁₀(e^x)

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

77. y = log₃(((x + 1)/(x − 1))^(ln 3))

Vehicular stopping distance Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is s = 1.1v + 0.054v², where s is measured in ft and v in mph. The linear term 1.1v models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term 0.054v² models the additional braking distance once they are applied. Find ds/dv at v = 35 and v = 70 mph, and interpret the meaning of the derivative.

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

63. y = x^π"

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

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

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

11. y = 5x^(3.6)

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.

Find the indicated derivative.

$f(x)=6x^4-4^3$

$f^{\(\prime\)}\(\left\)(x\(\right\))=$

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line has slope -1/2.

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

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

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

a. 6ƒ(x) - g(x), x = 1

Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.

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

f(s) = √s/4

Find and simplify the derivative of the following functions.

f(x) = √(e^2x + 8x²e^x +16x⁴) (Hint: Factor the function under the square root first.)

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

a. d/dx (f(x)+2g(x)) |x=3