Test 2

Question 1

(f of g) ‘

Answer 1

( f of g)= f (g) + g (f) `

Question 2

(N) / (D) `

Answer 2

(DN- DN) / D^2

Question 3

The derivative of a constant is

Answer 3

0

Question 4

Power rule

Answer 4

nx^n-1

Question 5

Constant multiple Rule

Answer 5

d/dx cf(x) = c d/dx f(x)

Question 6

Derivative of a sum

Answer 6

d/dx f(x) + d/dx g(x)

Question 7

derivative of a difference

Answer 7

d/dx f(x) - g(x) = d/dx f(x) - d/dx g(x)

Question 8

When finding the equation of a tangent line at x = c with y’

Answer 8

Differentiate the y function and then plug in c for x to find slope, then put into point slope form y-y1 = m(x-x1)

Question 9

where tangent line is horizontal

Answer 9

m = 0

Question 10

if f(x) = a^x then f’(x) = f’ (0)a^x

Answer 10

says the rate of change of an exponential function is proportional to the function itself

Question 11

Definition of number e

Answer 11

such that lim h approaches 0 (e^h - 1)/h = 1

Question 12

d/dx e^x

Answer 12

e^x

Question 13

The normal line to the curve at P

Answer 13

The normal line at P is perpendicular to the tangent line at P (point of tangency) and slopes are opposite reciprocal)

Question 14

When finding equations of the tangent line and the normal line to a curve

Answer 14

1) find slope of tangent line by plugging in the x value given and putting it in point slope form, then switch m’s reciprocal to find equation of tangent line

Question 15

y =

Answer 15

mx +b

Question 16

lim theta approaches 0 of (sin theta)/theta =

Answer 16

1

Question 17

The product rule

Answer 17

FS’ + SF’

Question 18

lim theta approaches 0 (costheta - 1)/theta =

Answer 18

0

Question 19

Derivative of sin x

Answer 19

cos x

Question 20

Derivative of cos x

Answer 20

• sin x

Question 21

Derivative of tan x

Answer 21

sec^2 x

Question 22

Derivative of csc x

Answer 22

-csc x cot x

Question 23

Derivative of sec x

Answer 23

sec x tan x

Question 24

Derivative of cot x

Answer 24

-csc^2 x

Question 25

The chain rule

Answer 25

F ‘ (x) = f ‘ (g(x)) times g ‘ (x). Basically, derivative of the outside function times the derivative of the inside function

Question 26

d/dx (a^x) =

Answer 26

a^x ln a

Question 27

Implicit differentiation

Answer 27

We differentiate both sides of the equation with respect to our independent variable x, bearing in mind that y is implicitly definied as a fuction of x

Question 28

Derivative of (sin^-1 x)

Answer 28

1/ (sqrt of 1 -x^2)

Question 29

Derivative of (cos^-1 x)

Answer 29

-1/ (sqrt of 1-x^2

Question 30

Derivative of (csc^-1 x)

Answer 30

-1/ x(sqrt of x^2 -1)

Question 31

Derivative of (sec^-1 x)

Answer 31

1/ x(sqrt of x^2-1)

Question 32

Derivative of (cot^-1 x)

Answer 32

-1/ 1 + x^2

Question 33

d/dx (log a x)

Answer 33

1/ x ln a

Question 34

d/dx (ln x)

Answer 34

1/x

Question 35

d/dx (log a u)

Answer 35

1/u ln a times u’

Question 36

d/dx (ln u)

Answer 36

1/ u(x) times u’ (x)

Question 37

d/dx ln |x|

Answer 37

1/x

Question 38

Exponential growth

Answer 38

if K > 0

Question 39

Exponential decay

Answer 39

if K < 0

Question 40

Differential equation for the Law of Natural growth

Answer 40

K is the relative growth rate, C is initial value, dy/dt = Ce^kt

Question 41

Newton’s Law of Cooling

Answer 41

T - Ts = Ce^kt, T(t) = Ts + Ce^kt, C = T - Ts

Question 42

d/dx (sinh x)

Answer 42

cosh x

Question 43

d/dx (cosh x)

Answer 43

sinh x

Question 44

exponential growth

Answer 44

dy/dx = Ce^kt

Question 45

Newton’s Law of cooling

Answer 45

T-Ts = Ce^kt

Question 46

Sinh x

Answer 46

e^x - e^-x
/2

Question 47

cosh x

Answer 47

e^x + e^-x
/2

Question 48

d/dx sinh x

Answer 48

cosh x

Question 49

d/dx cosh x

Answer 49

sinh x

Question 50

Linearization

Answer 50

L(x) = f(a) + f(a) (x-a)