Differentiation Rules (Chapter 3)

Question 1

What is the d/dx of e^x?

Answer 1

e^x

Question 2

What is the d/dx of cos x?

Answer 2

-sin x

Question 3

How do you find the derivative of f at x?

Answer 3

Lim as delta x -> 0: f(x + delta x) - f(x) / delta x

Question 3

What is the d/dx of e^u?

Answer 3

e^uu’

Question 4

What is the d/dx of ln x?

Answer 4

1 / x

Question 4

What is the d/dx of log_ax?

Answer 4

1 / (ln a) * x

Question 4

What is the d/dx of cos^-1(u)?

Answer 4

-u’ / sqrt(1-u²)

Question 5

What is the d/dx of sec^-1(u)?

Answer 5

u’ / |u| * sqrt (u²-1)

Question 5

What is the d/dx of |u|?

Answer 5

u / |u| * (u’)

Question 6

What is the d/dx of tan x?

Answer 6

sec²x

Question 7

What is the d/dx of a^x?

Answer 7

(ln a) * a^x

Question 7

What is the formula for volume of a conic?

Answer 7

1/3(pi)r²h

Question 10

What is the quotient rule ([d/dx] u / v)?

Answer 10

vu’ - uv’ / v²

Question 11

What is the alternate definition of a limit? How can this be used to determine differentiability with absolute value?

Answer 11

f(x) - f(x) / x - c

Evaluate the limit at c+ and c-. If they differ, the function is not differentiable.

Question 11

What is the d/dx of a^x?

Answer 11

(ln a) a^x

Question 13

What is the chain rule ([d/dx] f(u))?

Answer 13

f’(u) * u’ for each function in the expression.

Question 14

What is the d/dx of csc^-1(u)?

Answer 14

-u’ / |u| sqrt(u² - 1)

Question 15

What is the d/dx of a^u?

Answer 15

(ln a) a^u * u’

Question 17

What is the d/dx of sin x?

Answer 17

cos x

Question 18

What is the product rule ([d/dx] u * v)?

Answer 18

uv’ + vu’

Question 20

What is the simple power rule ([d/dx] xⁿ)?

Answer 20

For xⁿ, nx^n-1.

For x¹, 1.

Question 21

What is the derivative of a constant?

Answer 21

0

Question 22

What is the d/dx of sec x?

Answer 22

sec x tan x

Question 23

What is the constant multiple rule ([d/dx] c*u)?

Answer 23

c * u’

Question 24

What is the d/dx of csc x?

Answer 24

-csc x cot x

Question 25

What is the d/dx of sin^-1(u)?

Answer 25

u’ / sqrt(1 - u²)

Question 26

What is the d/dx of cot^-1(u)?

Answer 26

-u’ / 1 + u²

Question 27

What is the d/dx of cot x?

Answer 27

-csc²x

Question 28

What is the d/dx of log_ax?

Answer 28

1 / (ln a) x

Question 29

What is the sum / difference rule ([d/dx] u + v / u - v?)

Answer 29

u’ + v’ / u’ - v’

Question 30

What is the d/dx of tan^-1(u)?

Answer 30

u’ / 1 + u²

Question 32

What is the General Power Rule ([d/dx] un)?

Answer 32

nu^n-1 * u’

Question 33

What is the d/dx of ln u?

Answer 33

u’ / u

Question 33

What is the d/dx of log_au?

Answer 33

u’ / (ln a) u

Question 35

What would you use to find the d/dx of (x-2)^2 / sqrt(x^2 + 1)?

Answer 35

Logarithmic differentiation. Take Ln of both sides. Use ln u for ln y. Use Ln division rule Ln (x/y) Ln x - Ln y.

Question 36

How do you find the derivative of an inverse function?

Answer 36

It is the reciprocal of f’(g(x)) where g(x) is the inverse function.

Question 37

A conical tank with vertex down is 20 feet across the top and 16 feet deep. How can we find:

a. Water is flowing into the tank at a rate of 5 cubic feet per minute. How can we find the rate of change of the depth of the water when the water is 4 feet deep?
b. If the water is rising at a rate of 2/3 feet per minute when h = 8, how can we determine the rate at which water is being pumped into the tank?

Answer 37

Given 1/3 pi r^2 h:

a. Using similar triangles, find r = some value of h and back subtitute into the above formula (making sure to square and combine substituted h into h already present).

r/h = 10/16, 16r = 10h, r=10/16h = 5/8h

1/3 pi (5/8h)^2 h = 1/3 pi 25/64 h^3

Then, find derivative of modified formula, being sure to use implicit differentiation for h. This is dv/dt.

dv/dt = 75/192 * pi * h^2 * h’ = 25/64 * pi * h^2 * h’

Equate dv/dt with modified formula, plug in h (4) and solve for h’.

5 = 25/64 * pi * 4^2 * h’; h’ = 4/5pi ft/min

Question 38

A conical tank with vertex down is 20 feet across the top and 16 feet deep. How can we find:

b. If the water is rising at a rate of 2/3 feet per minute when h = 8, how can we determine the rate at which water is being pumped into the tank?

Answer 38

Using the formula already determined from item a, (dv/dt = 75/192 * pi * h^2 * dh/dt), plug in dh/dt (2/3) and h.

Question 39

What is Newton’s method?

Answer 39

1. Graph function.
2. Approximate Zeros visually by guessing a value of x close to the zero.
3. Draw a table with 5 columns:

n | x_n | f(x_n) | f‘(x_n) | x_n - f(x_n) / f’(x_n)

4. Compute all values. For row 1, x_n is your guessed value. For subsequent rows, x_n is the value from column 5 in the prior row.