Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

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

$\sqrt[3]{x^2+9}$

Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

y = (x² - 2ax + a²) / (x - a); a is a constant.

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

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

y = 4^-x sin x

Tangent lines Find an equation of the line tangent to the graph of f at the given point.

f(x) = sin⁻¹(x/4); (2,π/6)

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).

Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).

​​​​​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)

A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100xy.

9–61. Evaluate and simplify y'.

y = (2x−3)x^3/2

Evaluate the derivative of the following functions.

f(u) = csc^-1 (2u + 1)

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.

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

f(x) = x^In x

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

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

f(s) = √s/4

Find the derivative of the following functions.

y = e^-x sin x

Matching heights A stone is thrown with an initial velocity of 32 ft/s from the edge of a bridge that is 48 ft above the ground. The height of this stone above the ground t seconds after it is thrown is f(t) = −16t²+32t+48 . If a second stone is thrown from the ground, then its height above the ground after t seconds is given by g(t) = −16t²+v0t, where v0 is the initial velocity of the second stone. Determine the value of v0 such that both stones reach the same high point.

73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>

Exercise 46

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|

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

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

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

51–56. Second derivatives Find d²y/dx².

2x²+y² = 4

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = 3x² + 5e^x

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

sin x+sin y=y

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

f(x) = 2^x/2^x+1

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)

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

f(x) = x⁸cos³ x / √x-1

Find the derivative of the following functions.

y = cos x/sin x + 1

Find y'' for the following functions.

y = cos θ sin θ