Partial Derivative Examples

Business Calculus · Axiom Academy

EXAMPLE Partial Derivative Examples Step-by-step solutions for various function types Find all first-order partial derivatives: Treat y as a constant and differentiate with respect to x: The term is constant (derivative = 0), and in , y acts as a coefficient. Treat x as a constant and differentiate with respect to y: The term is constant (derivative = 0), and in , x acts as a coefficient. Use the chain rule. Let u = 2x + 3y, then : When differentiating , multiply by the partial derivative of the exponent. When differentiating with respect to y, is constant: Apply quotient rule with respect to y: 6 Business Application: Cobb-Douglas A production function is given by where L = labor and K = capital. Find the marginal productivities. Treat K as a constant. Use the power rule on L: Treat L as a constant. Use the power rule on K: MP L ≈ 27.1 means one additional unit of labor increases output by about 27.1 units. MP K ≈ 36.1 means one additional unit of capital increases output by about 36.1 units. Based on the example above, which approach is correct?

This is the written version of the interactive lesson above. See the full Business Calculus course.