3. Techniques of Differentiation

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(s) = 4s³+3s; a= -3, -1

Use the table to find the following derivatives.

<IMAGE>

d/dx (f(x) + g(x)) ∣x=1

Additional Applications

Bacterium population

When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10⁶ + 10⁴t − 10³t². Find the growth rates at

a. t = 0 hours.

b. t = 5 hours.

c. t = 10 hours.

{Use of Tech} Equations of tangent lines

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = −3x²+2; a=1

Find the indicated derivative.

$d/dt(5-7t^{-3})$

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

Airplane takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10/9)t², where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km/h. How long will it take to become airborne, and what distance will it travel in that time?

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (5f(x)+3g(x)) |x=1

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

g(t) = 6√t

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = g(x) at x = 3.

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?

Find the indicated derivative.

$y=\frac{5}{2}{x}^5+2x^3+6x-\frac{\surd 3}{4}$

$y^{\prime }=$

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 3t⁴; a= -2, 2

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

59. y = 2^x

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