Partial vs total derivatives in multivariable calculus

Question

In multivariable calculus, I see both partial derivatives $\frac{\partial f}{\partial x}$ and total derivatives $\frac{df}{dx}$ . When should I use each?

For a function $f(x, y)$ where $y = y(x)$ , the chain rule gives:

\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}

What does this mean intuitively? Can someone provide a concrete example where mixing them up leads to wrong results?

James Chen · Accepted Answer

The key distinction: - Partial derivative ∂ f ∂ x holds all other independent variables constant. It tells you how f changes when you wiggle x while keeping y fixed. - Total derivative d f d x 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 = 3 x. Then ∂ f ∂ x = 2 x (treating y as constant). But d f d x = ∂ f ∂ x + ∂ f ∂ y d y d x = 2 x + 2 y ⋅ 3 = 2 x + 6 y = 2 x + 6 ( 3 x ) = 20 x. We can verify: substituting y = 3 x gives f ( x ) = x 2 + ( 3 x ) 2 = 10 x 2, and indeed d d x ( 10 x 2 ) = 20 x. 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).

Partial vs total derivatives in multivariable calculus

1 Answer

Partial vs total derivatives in multivariable calculus

1 Answer