Basics Flashcards
What is the constant function rule?
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
What is the constant factor rule?
If y = c(f(x)
Then dy/dx = cf’(x)
What is the power function rule?
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
What is the sum-difference rule?
If y = f(x) + or – g(x) + or – h(x)
Then dy/dx = f’(x) + or – g’(x) + or – h’(x)
What are the economic applications of differentiation?
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
What is the product rule?
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)
What is the relationship between the MR, AR and TR functions?
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.
What is the marginal revenue function?
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.
What is the total revenue function?
TR = AR x Q = f(Q) x Q
What is the quotient rule?
If y = f(x)/g(x)
Then dy/dx = [f’(x)g(x) – f(x)g’(x)]/[g(x)]⮝2
What is the relationship between the MC, AC and TC functions?
TC = C = C(Q) AC = C(Q)/Q MC = dAC/Dq
What is the chain rule?
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
When do you have a strictly increasing function?
If x1 > x2 => f(x1) > f (x2)
When do you have a strictly decreasing function?
If x1 > x2 => f(x1) < f (x2)
What is a monotonic function?
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.
How do you determine if a function is monotonic?
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).
What is the log function rule?
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
What are some complicated log examples?
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)
What is the exponent function rule?
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)
How do you get rid of a natural log?
Put the function to the power e.
What does a square root equal?
√x = x⮝1/2
1/x⮝1/2 = 1/√x = x⮝-1/2
How do you compute the premium?
U(EV - P) = EU
