3. Techniques of Differentiation

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

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

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

f(v) = v¹⁰⁰+e^v+10

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

f(s) = 2√s-1; a=25

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

(g^-1)'(7)

{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = −3x² + 2; a=1

Find the indicated derivative.

$d/dt(-5x)$

{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = e^x; a = ln 3

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂0 e^x-1 / x

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?

Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).

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

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂1 x¹⁰⁰-1 / x-1

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line is horizontal.

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

h(t) = t²/2 + 1

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 = h(x) at x = 3.