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

y = 10^In 2x

23–51. Calculating derivatives Find the derivative of the following functions.

y = sin x / 1 + cos x

{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?

f(x) = sin x on [−π,π]

Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Find f′(7).

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

y = 4^-x sin x

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 derivative of the following functions.

y = In √x⁴+x²

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x²(3y²−2y³) = 4

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

27–76. Calculate the derivative of the following functions.

$y=\sqrt{10x+1}$

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

51–56. Second derivatives Find d²y/dx².

x⁴+y⁴ = 64

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

f(x) = 1/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>

Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w'(15) = 1.75.

Find the derivative of the following functions.

y = cos x/sin x + 1

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

P = 40/1+2^-t

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

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

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

f(x) = (1+ 1/x)^x

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

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

f(x) = (x+1)¹⁰ / (2x-4)⁸

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.

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

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

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.)

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

y = In (x³+1)^π

Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. <MATCH A-D IMAGE>