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What is the formula for the total differential of a function z=f(x,y)z = f(x,y)?

The total differential of a function z=f(x,y)z = f(x,y) is defined as the sum of the partial derivative of f with respect to x multiplied by dx, and the partial derivative of f with respect to y multiplied by dy. The formula is dz = (∂f/∂x)dx + (∂f/∂y)dy.

Conditions

  • z is a function of two variables, x and y.
  • The partial derivatives ∂f/xf/x and ∂f/yf/y exist.

Reasoning, step by step

  1. Identify the function z=f(x,y)z = f(x,y).
  2. Compute the partial derivative of f with respect to x, treating y as a constant.
  3. Compute the partial derivative of f with respect to y, treating x as a constant.
  4. Multiply the first partial derivative by dx and the second by dy.
  5. Add the two terms together to form the total differential dz.

Example

The instructor writes the definition on the whiteboard: the total differential dz is equal to the partial derivative of f with respect to x times dx, plus the partial derivative of f with respect to y times dy.

Common misconceptions

  • Confusing the total differential with the total derivative, though the instructor notes some people use the terms interchangeably.
  • Forgetting to treat the other variable as a constant when computing partial derivatives.

Watch the explanation

0:15 – 0:55Watch this moment ↗

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