Lecture
DIFFERENTIATION

1. Rules of differentiation

1. Functions of one variable
   • Constant function: y = f(x) = a. Obtain dy/dx.
   • Power function: f(x) = axⁿ, where a and n are constants. Obtain f´(x).
   • Note: n can be an integer or a fraction.

2. Example
   • f(x) = 32x^1/2
   • Obtain f´(x).
   • Obtain f´(25).

3. Two or more functions of one variable

   Consider the functions u(x) and v(x).

   The Sum-Difference rule

   • y = f(x) = u(x) ± v(x)
   • f´(x) = u´(x) ± v´(x)

   Product rule

   • y = u(x)v(x)
   • dy/dx = u(x)v´(x) + v(x)u´(x)

   Quotient rule

   • y = u(x)/v(x)
   • dy/dx = [v(x)u´(x) - u(x) v´(x)]/[v(x)]²

4. Examples

   1. y = (ax² + b)(cx³)

   • Obtain dy/dx.

   2. y = x/(1+x)

   • Obtain dy/dx.
   • Obtain dy/dx using the product rule.

2. Composite functions

1. Consider two functions: y = f(u); u = g(x)
   • Write y as a composite function of x:
     y = f(g(x))
   • Use the chain rule to obtain the derivative:
     dy/dx = f´(u)u´(x)

2. Examples
   • f(u) = 3u²; u = 4x + 2
   • y = 3(4x+2)²
   • Obtain dy/dx.

3. Function of more than one variable

1. Consider a function y = f(x₁, x₂,..., x_n)
   • y is the dependent variable
   • x₁, x₂,..., x_n are the independent variables.

2. Partial derivatives
   • What is the change in y due to a change in one of the independent variables, holding all others fixed?
   • Consider a change in x₁.
   • Obtain the difference quotient and apply limits.
   • Note: When taking partial derivatives, treat all other variables as constants; then apply the ordinary rules of differentiation.

3. Application: Marginal utility
   A consumer derives utility from the consumption of two goods X and Y. The consumer's utility function is: u(x,y) = x^1/3 y^2/3, where u represents utility, and x and y represent the amounts consumed of the two goods.
   • Obtain u₁.

     What does it represent?
     The marginal utility of good X, or the additional utility gained by an incremental increase in the consumption of good X, keeping Y constant.

   • If x = 8 and y = 27, what is the marginal utility of good Y?

4. Application: Marginal product
   A firm's production function is given by Y = F(K,L) = RK^a L^b, where y is output; K and L are inputs of capital and labor resp.; R, a and b are parameters.
   • Obtain the partial derivatives of Y w.r.t K and L.
   • What do F₁ and F₂ represent?

   Note: A Cobb-Douglas function is one where a + b = 1.

5. Partials of partials!
   • In general, the partial derivatives are themselves functions of the same independent variables. Thus, you can take their partial derivatives, thereby obtaining the second-order derivatives.

4. Exercises

1. Consider the utility function in 3(c).
   • Obtain the partial derivatives of u₁ and u₂ w.r.t x and y.
   • Show that u₁₁ and u₂₂ are negative.
   • Show that u₁₂ = u₂₁.

2. Consider a special case of the production function in 3(d).
   • Let b = 1 - a.
   • Obtain the partial derivatives of F₁ and F₂ w.r.t K and L.
   • Show that F₁₁ and F₂₂ are negative.
   • Show that F₁₂ = F₂₁.

   Note: The cross-partials are the same regardless of the order in which you perform the differentiation.

5. Total differential

1. Function of one variable
   • y = f(x)
   • Differential: dy = f'(x) dx
   • An infinitesimal change in x multiplied by the derivative, f'(x), yields the corresponding change in y.

2. Function of more than one variable
   • y = f(x₁, x₂,..., x_n)
   • Total differential: dy = f₁ dx₁ + f₂ dx₂ + ... + f_n dx_n

3. Example
   • Find the total differential, dz, of the function z = 4x² + 2y.
   • Answer: dz = 8x dx + 2 dy

6. Total derivative

1. Consider a function y = f(x,u), where u = g(x).

   What is the total derivative dy/dx?

   • Obtain the total differential of y:
     dy = f₁ dx + f₂ du
   • Divide both sides by the differential dx:
     dy/dx = f₁ + f₂ du/dx = f₁ + f₂ g'(x)

2. Example
   • y = 2x + u; u = 3x²
   • Find the total derivative dy/dx.

7. Implicit functions

1. y is not written explicitly as a function of x.
   f(x,y) = 0
   • f₁dx + f₂dy = 0

   • dy/dx = -f₁ / f₂

   Note: What condition must be met to obtain the derivative above?

2. Examples

   1. x² + y² = 16

   • Totally differentiate the equation:
     2x dx + 2y dy = 0
   • Obtain dy/dx = -x/y.
     [Note: Provided y does not equal 0.]

   2. y³x² = 1

   • Obtain dy/dx.