Unit 3: Differential Calculus - 1 (Single Variable Case)

1. Basic Rules of differentiation

Definition: The derivative of a function y = f(x), denoted as f'(x), dy/dx, or y', measures the instantaneous rate of change of y with respect to x.

Geometrically, the derivative at a point is the slope of the tangent line to the function's graph at that point.

Basic Rules of Differentiation:

Advanced Rules:

• Product Rule: For f(x) = u(x) * v(x) f'(x) = u'(x)v(x) + u(x)v'(x)
  Mnemonic: "First d-Second, plus Second d-First" (where d- means derivative of)
• Quotient Rule: For f(x) = u(x) / v(x) f'(x) = [ u'(x)v(x) - u(x)v'(x) ] / [ v(x) ]²
  Mnemonic: "Low d-High, minus High d-Low, all over Low-squared"
• Chain Rule: For a composite function f(x) = g(h(x)) f'(x) = g'(h(x)) * h'(x)
  Mnemonic: "Derivative of the outside (with the inside left alone), times the derivative of the inside."
  Example: y = (x² + 1)³
  Outside = (...)³, Inside = (x² + 1)
  dy/dx = 3(x² + 1)² * (2x) = 6x(x² + 1)²

2. Second and higher order derivative

A second derivative is simply the derivative of the first derivative. It measures the rate of change of the rate of change (i.e., acceleration).

• First Derivative: f'(x) or dy/dx
• Second Derivative: f''(x) or d²y/dx²
• Third Derivative: f'''(x) or d³y/dx³

What the Second Derivative Tells Us:

The sign of the second derivative tells us about the concavity of the function's graph.

• If f''(x) > 0 on an interval, the graph is Concave Up (like a "U"). The slope (f') is increasing.
• If f''(x) < 0 on an interval, the graph is Concave Down (like an "∩"). The slope (f') is decreasing.

A Point of Inflection is a point on the graph where the concavity changes (from up to down, or down to up). This occurs when f''(x) = 0 or is undefined, *and* the sign of f'' changes.

3. Convex and concave function

In economics, the terms "convex" and "concave" are often used in a slightly different (but related) way, especially with utility and production functions.

• Concave Function: A function where the line segment connecting any two points on the graph lies below the graph.
  • Test: f''(x) ≤ 0 (Concave Down)
  • Economic Example: A standard Production Function (like Q = √L) is concave. This shows diminishing marginal returns. The first worker adds 10 units, the second adds 8, the third adds 6... the slope is decreasing.
• Convex Function: A function where the line segment connecting any two points on the graph lies above the graph.
  • Test: f''(x) ≥ 0 (Concave Up)
  • Economic Example: A standard Total Cost (TC) function (after a certain point) is convex. This shows increasing marginal costs. The cost of producing one more unit gets higher and higher as you produce more.

4. Optimisation problem for function of one variable

Optimization is the process of finding the maximum or minimum value of a function (e.g., maximizing profit or minimizing cost).

Steps for Optimization:

1. Find Critical Points:

   These are the potential locations for a max or min. They occur where the tangent line is horizontal (slope is zero).

   First-Order Condition (FOC): Find all values of x where f'(x) = 0 or f'(x) is undefined.

2. Test the Critical Points (The "Second-Order Condition"):

   Once you have a critical point c (where f'(c) = 0), use the Second Derivative Test to see if it's a max, min, or neither.

   Second-Order Condition (SOC):

   • If f''(c) < 0 (concave down), you have a Local Maximum at x = c.
   • If f''(c) > 0 (concave up), you have a Local Minimum at x = c.
   • If f''(c) = 0, the test is inconclusive. (It could be an inflection point).

5. Economic application of differentiation

The single most important application of derivatives in economics is the concept of "marginal" change.

A. Marginal Concepts

If you have a Total function, its derivative is the Marginal function.

• Marginal Cost (MC):
  If TC(Q) = Total Cost, then MC(Q) = d(TC)/dQ.
  MC is the approximate cost of producing one more unit.
• Marginal Revenue (MR):
  If TR(Q) = Total Revenue, then MR(Q) = d(TR)/dQ.
  MR is the approximate revenue from selling one more unit.
• Marginal Product (MP):
  If TP(L) = Total Product (as a function of Labor), then MP_L = d(TP)/dL.
  MP_L is the output produced by one more unit of labor.

B. Profit Maximization

This is the central optimization problem for a firm.
Profit (π) = Total Revenue (TR) - Total Cost (TC)
π(Q) = TR(Q) - TC(Q)

Step 1: First-Order Condition (FOC)
To maximize profit, take the derivative with respect to Q and set it to zero.
dπ/dQ = d(TR)/dQ - d(TC)/dQ = 0
d(TR)/dQ = d(TC)/dQ

MR(Q) = MC(Q)

This is the famous profit-maximization rule: A firm maximizes profit at the quantity (Q) where marginal revenue equals marginal cost.

Step 2: Second-Order Condition (SOC)
To ensure this is a maximum (and not a minimum), the second derivative must be negative.
d²π/dQ² < 0
d(MR)/dQ - d(MC)/dQ < 0

d(MR)/dQ < d(MC)/dQ

This means that at the profit-maximizing quantity, the slope of the MC curve must be greater than the slope of the MR curve. (The MC curve must cut the MR curve from below).

C. Elasticity

Elasticity measures the percentage change in one variable in response to a 1% change in another. The derivative is a key part of its formula.

Price Elasticity of Demand (E_d):
E_d = (% Change in Q) / (% Change in P)

E_d = (dQ/dP) * (P/Q)

• (dQ/dP) is the derivative of the demand function Q = f(P).
• (P/Q) is the ratio of price to quantity at a specific point on the curve.