1. What is the Total Derivative?
The total derivative of a function expresses how the function changes with respect to changes in all of its variables, considering both direct and indirect dependencies.
If a function z = f(x, y), and x and y are both functions of another variable t, then the total derivative of z with respect to t is:
dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
2. Example
Let z = x²y, where x = t² and y = sin(t). Then:
- ∂z/∂x = 2xy
- ∂z/∂y = x²
- dx/dt = 2t
- dy/dt = cos(t)
Substituting into the total derivative formula:
dz/dt = (2xy)(2t) + (x²)(cos(t))
3. Geometric Interpretation
The total derivative represents the rate of change of the function along a curve or path in its domain. It captures the effect of all variables varying together.