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).

Use the graph of f(x)=|x| to find f′(x).

​​​​​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

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

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

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

Find the derivative of the following functions.

y = In |sin x|

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

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

e^y-e^x = C, where C is constant

Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>

d/dx (xg(x)) | x=2

Evaluate the derivative of the following functions.

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

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

Find the derivative of the following functions.

y = In(cos² x)

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