What is the difference between partial derivatives and total derivatives in multivariable calculus?

The key distinction: - **Partial derivative** $\frac{\partial f}{\partial x}$ holds all other independent variables constant. It tells you how $f$ changes when you wiggle $x$ while keeping $y$ fixed. - **Total derivative** $\frac{df}{dx}$ accounts for all paths through which changing $x$ affects $f$, including indirect paths via other variables that depend on $x$. **Concrete example:** Suppose $f(x, y) = x^2 + y^2$ and $y = 3x$. Then $\frac{\partial f}{\partial x} = 2x$ (treating $y$ as constant). But $\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx} = 2x + 2y \cdot 3 = 2x + 6y = 2x + 6(3x) = 20x$. We can verify: substituting $y = 3x$ gives $f(x) = x^2 + (3x)^2 = 10x^2$, and indeed $\frac{d}{dx}(10x^2) = 20x$. **When to use which:** Use partial derivatives for multivariate functions where variables are truly independent. Use total derivatives when variables are related (e.g., in physics along a path, or in the chain rule for composite functions).