Basics

Question 1

What is the constant function rule?

Answer 1

If y = f(x) = c (where c is a constant)
Then dy/dx = f’(x) = 0

e.g. if y = f(x) = 7, then dy/dx = f’(x) = 0

Question 2

What is the constant factor rule?

Answer 2

If y = c(f(x)

Then dy/dx = cf’(x)

Question 3

What is the power function rule?

Answer 3

If y = f(x) = x⮝2
Then dy/dx = f’(x) = nx⮝n-1

e.g.
if y = f(X) = 3x⮝3, then dy/dx = 3(3x⮝2) = 9x⮝2

Question 4

What is the sum-difference rule?

Answer 4

If y = f(x) + or – g(x) + or – h(x)

Then dy/dx = f’(x) + or – g’(x) + or – h’(x)

Question 5

What are the economic applications of differentiation?

Answer 5

In general, if our original functions represents a total function (e.g. total cost, total revenue, consumption function, etc.), then its derivative is its marginal function (e.g. marginal cost, marginal revenue, marginal propensity to consume, etc.)

Examples:
If total cost is: C(Q) = Q⮝3 + 4Q⮝2 + 10 Q + 75
Then the marginal cost is given by: MC = dC/dQ = 3Q⮝2 + 8Q + 10

Question 6

What is the product rule?

Answer 6

If y = f(x)g(x)
Then dy/dx = d/dx[f(x)g(x)]
= f(x) d/dx [g(x)] + g(x) d/dx [f(x)]
= f(x)g’(x) + g(x) + f’(x)

Question 7

What is the relationship between the MR, AR and TR functions?

Answer 7

Average revenue is a function of output: AR = f(Q)

Recall that: AR = TR/Q = PQ/Q = P, so P = f(Q), i.e. the average revenue curve is the inverse of the demand curve.

Question 8

What is the marginal revenue function?

Answer 8

MR = dTR/dQ = f’(Q)Q + f(Q)

Marginal revenue is given by the slope of the total revenue curve, so we simply find the derivative of the total revenue function.

Question 9

What is the total revenue function?

Answer 9

TR = AR x Q = f(Q) x Q

Question 10

What is the quotient rule?

Answer 10

If y = f(x)/g(x)

Then dy/dx = [f’(x)g(x) – f(x)g’(x)]/[g(x)]⮝2

Question 11

What is the relationship between the MC, AC and TC functions?

Answer 11

TC = C = C(Q)
AC = C(Q)/Q
MC = dAC/Dq

Question 12

What is the chain rule?

Answer 12

This is useful when we have a composite function.

If f(x) = p[q(x)]
Then f’(x) = p’[q(x)]q’(x)

OR
Let u = q(x) and y = p(u) so: y = p[q(x)] = f(x). Then: dy/dx = dy/du x du/dx

Question 13

When do you have a strictly increasing function?

Answer 13

If x1 > x2 => f(x1) > f (x2)

Question 14

When do you have a strictly decreasing function?

Answer 14

If x1 > x2 => f(x1) < f (x2)

Question 15

What is a monotonic function?

Answer 15

This is a function or quantity that varies in such a way that it either never decreases or never increases (a function is monotonic if its first derivative does NOT change sign).

For strictly increasing or strictly decreasing functions, f is said to be a strictly monotonic function and an inverse function, f⮝-1, exists.

Question 16

How do you determine if a function is monotonic?

Answer 16

A practical way of determining whether a function f(x) is strictly monotonic is to check whether the derivative f’(x) is either always positive or always negative (not zero) for all values of x. So:

If f’(x) > 0, then f(x) is strictly increasing (upward sloping)
If f’(x) < 0, then f(X) is strictly decreasing (downward sloping).

Question 17

What is the log function rule?

Answer 17

If y = f(x) = logb(x)
Then dy/dx = f’(x) = 1/xln(b)

For natural logs (base e), the rule becomes:
If y = f(x) = ln(x)
Then dy/dx = f’(x) = 1/xln(e) = 1/x

e.g. y = f(x) = 5ln(x), then dy/dx = f’(x) = 5/x

Question 18

What are some complicated log examples?

Answer 18

If y = ln g(x)
Then dy/dx = g’(x)/g(x)

e. g. if y = ln (x⮝2 + 1), then dy/dx = g’(x)/g(x) = 2x/(x⮝2 +1)
e. g. if y = x⮝2 ln(4x + 2), then, dy/dx = (x⮝2)(4/(4x + 2)) + (2x)(ln(4x + 2)) = 2x⮝2/(2x + 1) + 2xln(4x+2)

Question 19

What is the exponent function rule?

Answer 19

If y = f(x) = e⮝x
Then dy/dx = f’(x) = e

If y = f(x) = e⮝g(x)
Then dy/dx = f’(x) = g’(x) x e⮝g(x)

Question 20

How do you get rid of a natural log?

Answer 20

Put the function to the power e.

Question 21

What does a square root equal?

Answer 21

√x = x⮝1/2

1/x⮝1/2 = 1/√x = x⮝-1/2

Question 22

How do you compute the premium?

Answer 22

U(EV - P) = EU