What is the difference between total and partial derivatives?

What is the difference between total and partial derivatives?

Total derivatives consider all variables in a function, while partial derivatives focus on one variable at a time.

Can you give me an example of each?

Sure! For a function z = f(x, y), the partial derivative with respect to x is ∂f/∂x, while the total derivative is dz/dt = ∂f/∂x * dx/dt + ∂f/∂y * dy/dt.

When would I use total derivatives instead of partial derivatives?

You'd use total derivatives when you need to understand how a function changes with respect to time or another variable that affects all inputs.

Are there any applications of total derivatives in real life?

Yes, total derivatives are used in physics for motion analysis, in economics for cost functions, and in engineering for system dynamics.