1. Let's start by stating the problem: We want to understand the concept of partial derivatives, especially in the context of management mathematics. 2. A partial derivative measures how a multivariable function changes as one variable changes, keeping the other variables constant. 3. The general formula for the partial derivative of a function $f(x,y,\ldots)$ with respect to $x$ is: $$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y,\ldots) - f(x,y,\ldots)}{h}$$ 4. Important rules: - Treat all other variables as constants when differentiating with respect to one variable. - Partial derivatives help analyze how changing one factor affects the outcome in models with multiple variables. 5. Example: Suppose a profit function $P(x,y) = 5x^2y + 3xy^2$, where $x$ and $y$ are quantities of two products. 6. To find the partial derivative of $P$ with respect to $x$, treat $y$ as constant: $$\frac{\partial P}{\partial x} = \frac{\partial}{\partial x} (5x^2y + 3xy^2) = 10xy + 3y^2$$ 7. To find the partial derivative of $P$ with respect to $y$, treat $x$ as constant: $$\frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (5x^2y + 3xy^2) = 5x^2 + 6xy$$ 8. Interpretation: $\frac{\partial P}{\partial x}$ shows how profit changes when we increase product $x$ quantity, keeping $y$ fixed. 9. Similarly, $\frac{\partial P}{\partial y}$ shows how profit changes when we increase product $y$ quantity, keeping $x$ fixed. 10. This helps managers decide which product to focus on to maximize profit by analyzing marginal effects. In summary, partial derivatives are essential in management mathematics to understand the sensitivity of outcomes to changes in individual variables in multivariable models.