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(3x + 1)⁴

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

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?

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

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

Derivatives Find and simplify the derivative of the following functions.

g(t) = 3t² + 6/t⁷

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

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

y = 10^In 2x

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).

y = (cos x) In cos²x

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

y = x⁵

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = x^In x

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

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

f(x) = 1/tan^−1(x²+4)

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

h (x) = x^√x; a = 4

Find an equation of the line tangent to the curve y = sin x at x = 0.

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

Con​​sider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

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

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

Derivatives Find and simplify the derivative of the following functions.

f(x) = 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?

Use implicit differentiation to find dy/dx.

e^xy = 2y

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

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).

y = 4 log₃(x²−1)

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

x = y²

Continuity of a piecewise funct​​ion Let g(x) = <matrix 2x1> For what values of a is g continuous?