Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Question 1

What is the derivative of a composite function?

Answer 1

The derivative of a composite function is found using the chain rule.

Question 2

Fill in the blank: The chain rule is used to differentiate __________ functions.

Answer 2

composite

Question 3

What is the general formula for the chain rule?

Answer 3

If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).

Question 4

What is implicit differentiation?

Answer 4

Implicit differentiation is a method used to find the derivative of functions with x’s and y’s scrambled.

Question 5

True or False: Implicit differentiation requires that y be isolated on one side of the equation.

Answer 5

False

Question 6

When using implicit differentiation, what must you remember to apply when differentiating y?

Answer 6

You must apply the chain rule and y’

Question 7

Derivative of the inverse function

Answer 7

If g is the incerse of f, then g’(x) = 1/f’(g(x))

Question 8

True or False: If f(x) is a one-to-one function, then it has an inverse function.

Answer 8

True

Question 9

What is the relationship between a function and its inverse regarding their derivatives?

Answer 9

The derivative of the inverse function is the reciprocal of the derivative of the function

Question 10

What is the first step in finding the derivative of a composite function?

Answer 10

Identify the outer and inner functions.

Question 11

What does the notation dy/dx represent?

Answer 11

It represents the derivative of y with respect to x.

Question 12

True or False: The derivative of a constant function is zero.

Answer 12

True

Question 13

Fill in the blank: The derivative of ln(x) is __________.

Answer 13

1/x

Question 14

If h(x) = u(x)v(x), what is h’(x) using the product rule?

Answer 14

h’(x) = u’(x)v(x) + u(x)v’(x).

Question 15

What is the formula for the quotient rule?

Answer 15

If h(x) = u(x)/v(x), then h’(x) = (vu’-uv’)/v^2.

Question 16

What does the term ‘one-to-one function’ mean?

Answer 16

A function is one-to-one if it never takes the same value twice.

Question 17

What is the derivative of f(x) = cos(x^2)?

Answer 17

-2xsin(x^2)

Question 18

What is the relationship between the graphs of a function and its inverse concerning their slopes?

Answer 18

The slopes are reciprocals of each other at corresponding points.

Question 19

Steps to find the inverse of f(x)

Answer 19

1. Replace f(x) with y .
2. Swap x with y
3. Rearrange the function algebracally equal to y
4. Finally, replace y with f^−1(x)

Question 20

Definiton of a Derivative at a point “a”

Answer 20

f’(a) = lim h->0 (f(a + h) - f(a)) / h

Question 21

Definition of a derivative as a function

Answer 21

f’(x) = lim h->0 (f(x - h) - f(x)) / h

Question 22

Whats the reciprocal?

Answer 22

1 divided by that number

Question 23

d/dx sin x =

Answer 23

cos x

Question 24

d/dx cos x =

Answer 24

-sin x

Question 25

d/dx tan x =

Answer 25

sec^2 x

Question 26

d/dc csc x =

Answer 26

(-csc x)(cot x)

Question 27

d/dx sec x =

Answer 27

(sec x)(tan x)

Question 28

d/dx cot x =

Answer 28

-csc^2 x

Question 29

d/dx a^x =

Answer 29

a^x ln a

Question 30

d/dx loga x =

Answer 30

1 / x ln a

Question 31

logarithmic differentiation

Answer 31

Sometimes, it can be helpful to take the log of both sides then use implicit differentiation to solve

Question 32

Solve y = (x^3 - x + 1)^4 using the chain rule

Answer 32

y = 4(x^3 - x +1)(3x^2 - 1)

Question 33

Derivative of a constant multiple

Answer 33

d/dx cu = c * d/dx u

Question 34

units of the derivative of particle motion are…

Answer 34

units of y / units of x

Question 35

If s(t) is the position function of an object, then s’(t) gives:

Answer 35

v(t) gives the object’s velocity

Question 36

If s(t) is the position function of an object, then s’‘(t) gives:

Answer 36

a(t) gives the object’s accelaration

Question 37

If s(t) is the position function of an object, then s’’‘(t) gives:

Answer 37

j(t) gives the object’s jerk

Question 38

Displacement

Answer 38

Overall change in position

Question 39

How much ground the object covered is…

Answer 39

total distance

Question 40

What does velocity tells you?

Answer 40

how fast you’re going and in what direction

Question 41

What does speed tells you?

Answer 41

How fast you’re going without regard to direction

Question 42

How to find speed?

Answer 42

Take the absolute value of velocity (v(t))

Question 43

Positive velocity is:

Answer 43

A particle moving forward/right on a horizontal line

Question 44

Negative velocity is:

Answer 44

A particle moving backwards/left on a horizontal line

Question 45

v(t) = 0 means

Answer 45

Particle is at rest

Question 46

Formula for avg. v(t) on [a,b]

Answer 46

(s(b) - s(a)) / b - a

Question 47

A particle is speeding up when…

Answer 47

velocity and acceleration have the same sign

Question 48

A particle is slowing down when…

Answer 48

velocity and acceleration have different signs

Question 49

Steps to create a sign chart for particle motion

Answer 49

1. Find Zeros, set function = 0
2. Mark Zeros on a Number Line
3. Substitute each test interval bwt zeros into the function
4. Determine if its + or - in that interval

Question 50

What info does a sign chart gives you about the motion of a particle?

Answer 50

helps analyze changes in direction and speed based on where there are zeros

Question 51

Formula for tangent line at x = a

Answer 51

y = f(a) + f’(a)(x - a)

Question 52

formula for the normal line at x = a

Answer 52

y = f(a) - (1/f’(a))(x - a)

Question 53

d/dx e^x =

Answer 53

e^x

Question 54

d/dx arcsin x =

Answer 54

1 / √(1-x^2)

Question 55

d/dx arccos x =

Answer 55

-1 / √(1-x^2)

Question 56

d/dx arctan x =

Answer 56

1 / (x^2 + 1)

Question 57

d/dx arccot x =

Answer 57

-1 / (x^2 + 1)

Question 58

d/dx arcsec x =

Answer 58

1 / (|x|•√(x^2 - 1))

Question 59

d/dx arccsc =

Answer 59

-1 / (|x|•√(x^2 - 1))