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

f(x) = x^In x

The line tangent to the graph of f at x=5 is y = 1/10x-2. Find d/dx (4f(x)) |x+5

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

Graph the following curves and determine the location of any vertical tangent lines.

a. x²+y² = 9

Angle to a particle (part 2) The figure in Exercise 81 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in the two answers. <IMAGE>

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

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

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

$y={(x^2+2x+7)}^8$

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

x = y²

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

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

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 = x² In x

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

The speed of sound (in m/s) in dry air is approximated the function v(T) = 331 + 0.6T, where T is the air temperature (in degrees Celsius). Evaluate v' (T) and interpret its meaning.

Derivatives Find and simplify the derivative of the following functions.

g(t) = 3t² + 6/t⁷

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

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(sec⁴x tan² x)

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

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

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

f(x) = tan¹⁰x / (5x+3)⁶

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

h(t) = t²/2 + 1

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

Find and simplify the derivative of the following functions.

f(x) = e^x(x³ − 3x² + 6x − 6)

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

f(x) = x⁸cos³ x / √x-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).

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.)

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>

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.)

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

y = In (x³+1)^π