Chapter 3

Question 1

What is the constant rule? (DxC=?)

Answer 1

The derivative of a constant is always 0. (DxC=0)

Question 2

What is the derivative of x? (DxX=?)

Answer 2

The derivative of x is 1. (DxX=1) (y=x)

Question 3

What is the power rule for derivatives? (DxX^n=?)

Answer 3

The derivative of x^n is nx^(n-1). (DxX^n=nX^(n-1))

Question 4

What is the sum/difference rule? ((f+g)prime=?)

Answer 4

The sum/difference of (f +/- g)prime is fprime +/-g prime.

Question 5

What is the derivative of a constant times a function of x? (DxCf(x)=?)

Answer 5

The constant times the derivative of the function of x. (DxCf(x)=C*fprime(x))

Question 6

What is the definition of the number e?

Answer 6

The number e is such that the lim h->0 (e^h-1)/h=1. (That the tangent slope of e^x at x=0 is 1)

Question 7

What is the product rule? ((fgl*m)prime)=?

Answer 7

((fglm)prime= fprimeglm+fgprimelm+fglprimem+fgl*mprime

Question 8

What is the quotient rule? (f/g)prime=?

Answer 8

(f/g)prime=(fprimeg-fgprime)/g^2

Question 9

DxSinx=?

Answer 9

Cosx

Question 10

DxCosx=?

Answer 10

-Sinx

Question 11

DxTanx=?

Answer 11

Sec^2x

Question 12

DxCscx=?

Answer 12

-CscxCotx

Question 13

DxSecx=?

Answer 13

SecxTanx

Question 14

DxCotx=?

Answer 14

-Csc^2x

Question 15

What is lim (theta)->0 Sin(theta)/(theta)?

Answer 15

lim (theta)->0 Sin(theta)/(theta)=1

Question 16

What is lim (theta)->0 (Cos(theta)-1)/(theta)?

Answer 16

lim (theta)->0 (Cos(theta)-1)/(theta)=0

Question 17

What is the Chain Rule? Dx(f o g)(x)=?

Answer 17

The Chain Rule states that the derivative of (f o g)(x) is f(prime)(g(x))(gprime(x) (the derivative of the outside function with the inside function intact and still inside it multiplied by the derivative of the inside)

Question 18

What is the derivative of e^x?

Answer 18

Dx e^x=e^x

Question 19

What is the derivative of e^f(x)?

Answer 19

Dx e^f(x)=e^f(x)*fprime(x) (The derivative of e to the function f(x) is e to the function f(x) multiplied by the derivative of the function f(x)) (This comes from the chain rule)

Question 20

What is the method for implicit differentiation? What do you use this for

Answer 20

Given an equation, find the prime (thinking of y(dx/dy) as f(x))(use the chain rule for substituting), and then solve for dx/dy. With the found equation, you can find the slope of the tangent line at a given point by plugging in values of the given point.

Question 21

DxSin^-1(x)=?

Answer 21

DxSin^-1(x)= 1/√(1-x^2)

Question 22

DxCos^-1(x)=?

Answer 22

DxCos^-1(x)= -1/√(1-x^2)

Question 23

DxSec^-1(x)=?

Answer 23

DxSec^-1(x)= 1/x√(x^2-1)

Question 24

DxCsc^-1(x)=?

Answer 24

DxCsc^-1(x)= -1/x√(x^2-1)

Question 25

DxCot^-1(x)=?

Answer 25

DxCot^-1(x)= -1/(x^2+1)

Question 26

DxTan^-1(x)=?

Answer 26

DxTan^-1(x)=1/(x^2+1)

Question 27

What is the Derivation of Log sub a of x? DxLogaX=?

Answer 27

The Derivation of the Log sub a of x is 1/xlna. DxLogaX= 1/((x)lna)

Question 28

What is the Derivation of the Natural log of x? DxLnx=?

Answer 28

The Derivation of the Natural log of x is 1/x. Dxlnx=1/x

Question 29

What is the Derivation of Log sub a of f(x)? DxLogaF(x)=?

Answer 29

The Derivation of Log sub a of f(x) is 1/(f(x) *lna)*fprime(x). DxlogaF(x)=1/(f(x)*lna)*fprime(x)

Question 30

What is the Derivation of the Natural log of f(x)? DxLnf(x)=?

Answer 30

The Derivation of the Natural log of f(x) is 1/f(x) *fprime(x). DxLnf(x)=1/f(x) *fprime(x)

Question 31

What is Logarithmic Differentiation? How do you use it? When do you use it?

Answer 31

Logarithmic Differentiation is using the properties of logarithms and Implicit Differentiation to find dy/dx. 1st take the ln of both sides, then simplify the Right Hand Side, then use Implicit Differentiation. If y is on the RHS, substitute the given y for it. You use it when there is a function x to the power of another function x.

Question 32

What is the equation for the area of the circle?

Answer 32

A=pi(r)^2

Question 33

How do you evaluate a limit at infinity? (Chapter 2 Review)

Answer 33

You multiply both the numerator and the denominator by 1/x^n if n is the highest power of x found in the denominator

Question 34

What is the equation for the volume of a sphere?

Answer 34

V=4/3pi(r)^3

Question 35

How do you calculate Average Rate of Change? (Chapter 2 Review)

Answer 35

Its just the slope formula ya dummy

Question 36

What is the equation for the surface area of a sphere?

Answer 36

4pi(r)^2

Question 37

What is the equation for the volume of a cone?

Answer 37

(pi(r)^2)*(h/3)

Question 38

How do you find theta using trig?

Answer 38

Soh-Cah-Toa, (if opposite side is 4 and adjacent side is 3, sin(theta)=4/3)

Question 39

What is the derivative of n^x?

Answer 39

Dx n^x=N^x*lnx*fprime(x)

Question 40

What is the Law of Cosines? (For not a right triangle)

Answer 40

c^2=a^2+b^2-2(a)(b)(CosC)

Question 41

What is the Law of Sines? (For not a right triangle)

Answer 41

SinA/a=SinB/b=SinC/c

Question 42

What is the definition of the "Normal Line" in relation to the tangent line?

Answer 42

The Normal Line is the line that is perpendicular to the tangent line at the point of tangency

Question 43

What is the equation for the area of a triangle?

Answer 43

1/2base*height