9–61. Evaluate and simplify y'.

y = tan (sin θ)

9–61. Evaluate and simplify y'.

y=sin √cos² x+1

66–71. Higher-order derivatives Find and simplify y''.

x + sin y = y

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot x)

9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

Evaluate and simplify y'.

xy⁴+x⁴y=1

Evaluate and simplify y'.

sin x cos(y−1) = 1/2

Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

y =√x³+x−1 at y=3

Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.

f(x) = x^-1/3

A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later? 

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>

Compute the average rate of population growth from 1950 to 1960. 

{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.

b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.

s(t) = -16t² + 128t + 192; a = 2

Use definition (1) (p. ​133​) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x² - 5x + 1; P(1,-7)

Use definition (1) (p. ​133​) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

An object dropped from rest falls d(t)=16t² feet in t seconds. Find d′(4).

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

Suppose the speed of a car approaching a stop sign is given by v (t) = (t-5)², for 0 ≤ t ≤ 5, where t is measured in seconds and v(t) is measured in meters per second. Find v′(3).

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim x🠂1) 3x²+4x-7 / x-1

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim x🠂2) 1/x+1 - 1/3 / x-2

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim h🠂0) (2+h)⁴-16 / h

Another way to ​​approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 2x + 1; P(0,1)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = -7x; P(-1,7)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 3x² - 4x; P(1, -1)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 8 - 2x²; P(0, 8)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = x² - 4; P(2, 0)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P (1,1)

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

a. Estimate L' (1.5) and state the physical meaning of this quantity.