Examples
The volume V of a cone depends on the cone's height h and its radius r according to the formula
The partial derivative of V with respect to r is
which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to h is
which represents the rate with which the volume changes if its height is varied and its radius is kept constant.
By contrast, the total derivative of V with respect to r and h are respectively
and
The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.
If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio k,
This gives the total derivative with respect to r:
Which simplifies to:
Similarly, the total derivative with respect to h is:
Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics, engineering, and other sciences and applied disciplines.
Read more about this topic: Partial Derivative
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