Partial Derivatives and Incremental Changes

Rate of Change

• The rate of change of z with respect to x is represented by ∂z/∂x.
• The total differential of z = f(x, y) is given by dz = (∂f/∂x)dx + (∂f/∂y)dy.
• The total differential gives an approximation of the change in the dependent variable given small changes in each of the independent variables.
• The total derivative represents the total change in the dependent variable due to changes in all the independent variables.
• The effect on z of a small change in x is given by the partial differential.
• The total differential is the sum of the partial differentials.
• The total derivative of z with respect to x is ∂z/∂x = ∂f/∂x + ∂f/∂y ⋅ dy/dx.

Partial Derivatives

• ∂f/∂x = 2x.

Total Derivative

• The total derivative of z with respect to t becomes ∂z/dt = ∂f/∂x ⋅ dx/dt + ∂f/∂y ⋅ dy/dt.
• The direct effect of x on z, plus the indirect effect of x on z through y is given by the total derivative.
• ∂f/∂x = x.

Derivative, Differential and Change

• The derivative refers to the rate of change of a function.
• A supply function is given by the equation P = aQ^2.
• Find the derivative of P, the differential of P, and the approximate change in P if Q is increased by 5%.

Total Differential, Partial Derivative, and Total Derivative

• The total differential of z is given by dz = (∂z/∂x)dx + (∂z/∂y)dy.
• The derivative of z is given by the partial derivative of z with respect to x and y.
• The formula for the total derivative is given by dz/dx = (∂z/∂x) + (∂z/∂y) * dy/dx.
• The differential refers to the actual change in the function that occurs when the independent variable is altered.

Example: Supply Function

• The derivative of P is dP/dQ = 6Q.
• The differential of P is dP = (6Q)dQ.
• The change in Q is ΔQ = 5% · Q.
• The approximate change in P is ΔP = (6Q)ΔQ.

Revenue Function

• The annual revenues (R) of Highple Consulting depends on the fees (F), promotions (P), and the number of marketers (M).
• The supply function is given by the equation P = aQ^2.