Editing Partial derivative (section)

===Geometry===
[[Image:Cone 3d.png|thumb|The volume of a cone depends on height and radius]]
The [[volume]] {{mvar|V}} of a [[cone (geometry)|cone]] depends on the cone's [[height]] {{mvar|h}} and its [[radius]] {{mvar|r}} according to the formula

<math display="block">V(r, h) = \frac{\pi r^2 h}{3}.</math>

The partial derivative of {{mvar|V}} with respect to {{mvar|r}} is

<math display="block">\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},</math>

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 {{mvar|h}} equals {{nowrap|<math display="inline">\frac{1}{3}\pi r^2</math>,}} 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|''total'' derivative]] of {{mvar|V}} with respect to {{mvar|r}} and {{mvar|h}} are respectively

<math display="block">\begin{align}
  \frac{dV}{dr} &= \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{dh}{dr}\,, \\
  \frac{dV}{dh} &= \overbrace{\frac{\pi r^2}{3}}^\frac{\partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{dr}{dh}\,.
\end{align}</math>

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 {{mvar|k}},

<math display="block">k = \frac{h}{r} = \frac{dh}{dr}.</math>

This gives the total derivative with respect to {{mvar|r}},

<math display="block">\frac{dV}{dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k\,,</math>

which simplifies to

<math display="block">\frac{dV}{dr} = k \pi r^2,</math>

Similarly, the total derivative with respect to {{mvar|h}} is

<math display="block">\frac{dV}{dh} = \pi r^2.</math>

The total derivative with respect to {{em|both}} {{mvar|r}} and {{mvar|h}} of the volume intended as scalar function of these two variables is given by the [[gradient]] vector 

<math display="block">\nabla V = \left(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}\right) = \left(\frac{2}{3}\pi rh, \frac{1}{3}\pi r^2\right).</math>