A **total derivative** of a multivariable function of several variables, each of which is a function of another argument, is the derivative of the function with respect to said argument. it is equal to the sum of the partial derivatives with respect to each variable times the derivative of that variable with respect to the independent variable. For example, given a function $ f(x,y,z) $ , and $ t $ with $ x,y,z $ being functions of $ t $ ,

- $ \frac{df}{dt}=\frac{\part f}{\part x}\cdot\frac{dx}{dt}+\frac{\part f}{\part y}\cdot\frac{dy}{dt}+\frac{\part f}{\part z}\cdot\frac{dz}{dt}+\frac{\part f}{\part t} $

The formula for a total derivative is a direct result of the chain rule.

Total derivatives are often used in related rates problems; for example, finding the rate of change of volume when two parameters are changing with time.

## Example

The radius and height of a cylinder are both $ 2cm $ . The radius is decreased at $ 1\frac{cm}{s} $ and the height is increasing at $ 2\frac{cm}{s} $ . What is the change in volume with respect to time at this instant?

The volume of a right circular cylinder is

- $ V=\pi r^2h $

We can take the total derivative of this with respect to time to get

- $ \frac{dV}{dt}=\frac{\part V}{\part r}\cdot\frac{dr}{dt}+\frac{\part V}{\part h}\frac{dh}{dt}=2\pi rh\cdot\frac{dr}{dt}+\pi r^2\cdot\frac{dh}{dt} $

- $ =2\pi(2)(2)(-1)+\pi(2)^2(2)=-8\pi+8\pi=0 $

At this moment, the volume of the cylinder is not changing.