Total Derivative — Definition, Formula & Examples

The total derivative captures the complete rate of change of a function with respect to a variable, accounting for all paths through which that variable influences the output — including indirect effects through other dependent variables.

For a differentiable function $f(x_1, x_2, \ldots, x_n)$ where each $x_i$ depends on a parameter $t$ , the total derivative of $f$ with respect to $t$ is $\frac{df}{dt} = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}$ . More generally, the total derivative of a function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$ at a point $\mathbf{a}$ is the unique linear map $D\mathbf{f}(\mathbf{a})$ such that $\lim_{\mathbf{h} \to \mathbf{0}} \frac{\|\mathbf{f}(\mathbf{a}+\mathbf{h}) - \mathbf{f}(\mathbf{a}) - D\mathbf{f}(\mathbf{a})\mathbf{h}\|}{\|\mathbf{h}\|} = 0$ .

Key Formula

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

Where:

$f$ = A function of x and y
$x, y$ = Intermediate variables, each depending on t
$t$ = The independent parameter

How It Works

When a function depends on several variables that are themselves functions of another variable, you cannot just take one partial derivative to find the overall rate of change. The total derivative applies the chain rule across every dependency. You compute each partial derivative of

f

, multiply it by the derivative of the corresponding intermediate variable, and sum the results. When the function maps

\mathbb{R}^n \to \mathbb{R}^m

, the total derivative at a point is represented by the Jacobian matrix.

Worked Example

Problem: Let

f(x, y) = x^2 y

where

x = 3t

and

y = t^2

. Find

\frac{df}{dt}

t = 1

Step 1: Compute the partial derivatives of f.

\frac{\partial f}{\partial x} = 2xy, \quad \frac{\partial f}{\partial y} = x^2

Step 2: Compute the derivatives of x and y with respect to t.

\frac{dx}{dt} = 3, \quad \frac{dy}{dt} = 2t

Step 3: Apply the total derivative formula and evaluate at

t = 1

(where

x = 3

y = 1

\frac{df}{dt} = (2xy)(3) + (x^2)(2t) = (2 \cdot 3 \cdot 1)(3) + (9)(2) = 18 + 18 = 36

Answer:

\frac{df}{dt}\big|_{t=1} = 36

Why It Matters

The total derivative is central to the multivariable chain rule, optimization with constraints, and differential geometry. In physics and engineering, it appears as the material derivative in fluid mechanics, tracking how quantities change along a moving particle's path.

Common Mistakes

Mistake: Confusing the total derivative with a partial derivative when intermediate variables depend on the same parameter.

Correction: A partial derivative holds other variables fixed. The total derivative accounts for all variables changing simultaneously — omitting any term gives an incomplete rate of change.