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

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

d. d/dx (f(x)³) |x=5

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

(g^-1)'(7)

Second derivatives Find d²y/dx² for the following functions.

y = x cos x²

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

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

Determine whether the following statements are true and give an explanation or counterexample.

d/dx((√2)^x) = x(√2)^{x - 1}

Determine whether the following statements are true and give an explanation or counterexample.

(4x+1)^{ln x} = x^ln(4x+1)

Second derivatives Find d²y/dx² for the following functions.

y = √x²+2

Second derivatives Find d²y/dx² for the following functions.

y = e^-2x²

Derivatives by different methods

a. Calculate d/dx (x²+x)² using the Chain Rule. Simplify your answer.

Tangent lines Determine an equation of the line tangent to the graph of y=(x²−1)² / x³−6x−1 at the point (0,−1).

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)

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the 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.

y = f(x)g(x)

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)

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(1)

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

a. Find L′(1) and interpret the meaning of this value.

​Composition containing si​n​ x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

Product Rule for three functi​​ons Assume f, g, and h are differentiable at x.

a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(x)).

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955.

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Estimate the instantaneous rate of growth in 1985. 

Use the definition of the derivative to evaluate the following limits.

${\lim }_{h\to 0}{\frac{\ln (e^8+h)-8}{h}}_{}$

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

a. Determine the average and marginal costs for x = 3000 lawn mowers.

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

b. Interpret the meaning of your results in part (a).

Use the definition of the derivative to evaluate the following limits.

${\lim }_{x\to 2}{\frac{5^x-25}{x-2}}_{}$

A spherical balloon is inflated at a rate of 10 cm³/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?

Water flows into a conical tank at a rate of 2 ft³/min. If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.