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Problem 3.5.54
Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (csc x) = -csc x cot x
Problem 3.8.35
Use implicit differentiation to find dy/dx.
x3 = (x + y) / (x - y)
Problem 3.4.23
Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t
Problem 3.10.48
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) = 1/2x+8; (10,4)
Problem 3.5.59
Find y'' for the following functions.
y = ex sin x
Problem 3.8.8
5–8. Calculate dy/dx using implicit differentiation.
e^y-e^x = C, where C is constant
Problem 3.4.19
Derivatives Find and simplify the derivative of the following functions.
f(x) = 3x⁴(2x²−1)
Problem 3.9.19
Find the derivative of the following functions.
y = In |sin x|
Problem 3.10.13
Evaluate the derivative of the following functions.
f(x) = sin-1 2x
Problem 3.10.38
Evaluate the derivative of the following functions.
f(x) = sin(tan-1 (ln x))
Problem 3.4.36
Find and simplify the derivative of the following functions.
f(x) = ex(x3 − 3x2 + 6x − 6)
Problem 3.11.37
Water is drained out of an inverted cone that has the same dimensions as the cone depicted in Exercise 36. If the water level drops at 1 ft/min, at what rate is water (in ft³/min) draining from the tank when the water depth is 6 ft?
Problem 3.4.21
Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1
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 3.9.63
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)
Problem 3.4.59
Find and simplify the derivative of the following functions.
f(x) = √(e2x + 8x2ex +16x4) (Hint: Factor the function under the square root first.)
Problem 3.8.56
51–56. Second derivatives Find d²y/dx².
sin x + x²y =10
Problem 3.9.50
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.
g (x) = x^ In x; a = e
Problem 3.10.2
Find the slope of the line tangent to the graph of y = sin^−1 x at x=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 3.11.21
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 3.2.11
Use limits to find f' (x) if f(x) = 7x.
Problem 3.10.21
Evaluate the derivative of the following functions.
f(y) = tan-1 (2y2 - 4)
Problem 3.9.95
{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.
Problem 3.3.89
Calculator limits Use a calculator to approximate the following limits.
lim x🠂0 e^3x-1 / x
Problem 3.10.3
Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.
Problem 3.36
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
Problem 3.26
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
Problem 3.5.85
Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?
Problem 3.5.64
Find y'' for the following functions.
y = cos θ sin θ