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

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

67​–​78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.

f(x) = e^3x+1

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

h(x) = √x (√x-x³/²)

5–8. Calculate dy/dx using implicit differentiation.

x = y²

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)

Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

r(t) = (e^2t + 3e^t + 2) / (e^t + 2)

How are the derivatives of sin^−1 x and cos^−1 x related?

A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

Simplify the expression e^xln(x²+1).

15–48. Derivatives Find the derivative of the following functions.

P = 40/1+2^-t

Evaluate the derivative of the following functions.

f(x) = sec^-1 (ln x)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Find the derivative of the following functions.

y = cos x/sin x + 1

13​–​40.​ Evaluate the derivative of the following functions.

f(x) = 1/tan^−1(x²+4)

Given that f(1) = 5, f′(1) = 4, g(1) = 2, and g′(1) = 3 , find d/dx (f(x)g(x))∣ ∣x=1 and d/dx (f(x) / g(x)) ∣ x=1.

A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?

A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?

​​​​​47​–​56.​ Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=tan x; (1,π/4)

Find y'' for the following functions.

y = e^x sin x

Evaluate the derivative of the following functions.

f(x) = sin^-1 (e^-2x)

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

6x³+7y³ = 13xy

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

f(s) = 2√s-1; a=25

Derivatives Find and simplify the derivative of the following functions.

s(t) = t⁴/³ / e^t

The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?

Define the acceleration of an object moving in a straight line.

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