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

Find the derivative of the following functions.

y = x² (1 - In x²)

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

y = xe^y

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)

Find y'' for the following functions.

y = tan x

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?

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

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

Find y'' for the following functions.

y = x sin x

A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.) <IMAGE>

Calculator limits Use a calculator to approximate the following limits.

lim x🠂0 e^3x-1 / x

Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).

f(x) = 1/x

Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>

How is lim x🠂0 sin x/x used in this section?

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In 2x/(x² + 1)³

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

y = x⁵

Derivatives Find and simplify the derivative of the following functions.

h(w) = w⁵/³ / w⁵/³+1

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (x²+1)^x

​​​​​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)

Evaluate the derivative of the following functions.

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

Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)

A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft³/min) if the water level is dropping at 6 min/in?

Find the derivative of the following functions.

y = sin x + cos x

The graph of y =x^{ln x} has one horizontal tangent line. Find an equation for it.

Evaluate the derivative of the following functions.

f(y) = tan^-1 (2y² - 4)

Find the following higher-order derivatives.

dⁿ/dxⁿ (2^x)

Parabolic ​motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>

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

y = In (x³+1)^π

Find y'' for the following functions.

y = cos θ sin θ