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Problem 3.10.35
Evaluate the derivative of the following functions.
f(x) = sec-1 (ln x)
Problem 3.5.25
Find the derivative of the following functions.
y = e^-x sin x
Problem 3.2.33
Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2
Problem 3.5.13
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 7x) / 3x
Problem 3.4.15
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.
Problem 3.2.10
Use the graph of f(x)=|x| to find f′(x).
Problem 3.9.82
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x⁸cos³ x / √x-1
Problem 3.4.18
Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).
Problem 3.2.76
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
Problem 3.17
The line tangent to the graph of f at x=5 is y = 1/10x-2. Find d/dx (4f(x)) |x+5
Problem 3.10.31
Evaluate the derivative of the following functions.
f(u) = csc-1 (2u + 1)
Problem 3.9.22
Find the derivative of the following functions.
y = In √x⁴+x²
Problem 3.11.21
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.)
Problem 3.8.7
5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x
Problem 3.3.7
Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).
Problem 3.5.23
Find the derivative of the following functions.
y = sin x + cos x
Problem 3.10.9
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.
Problem 3.4.46
Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)
Problem 3.6.9
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.
Problem 3.9.66
63–74. Derivatives of logarithmic functions 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 = log₈ |tan x|
Problem 3.8.54
51–56. Second derivatives Find d²y/dx².
x⁴+y⁴ = 64
Problem 3.7.28
27–76. Calculate the derivative of the following functions.
Problem 3.8.52
51–56. Second derivatives Find d²y/dx².
2x²+y² = 4
Problem 3.3.70
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x2 + 5ex
Problem 3.8.27
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
sin x+sin y=y
Problem 3.5.42
23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x
Problem 3.5.21
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.
Problem 3.2.64
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y= √x; P(4, 2)
Problem 3.1.58
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
Problem 3.5.29
Find the derivative of the following functions.
y = cos x/sin x + 1