What is the Difference Between Partial and Total Derivative?

by Alexander FufaeV

The difference between a partial derivative $\frac{\partial f(x,y)}{\partial x}$ to $x$ and a total derivative $\frac{\text{d}f(x,y)}{\text{d}x}$ to $x$ is that in the partial derivative it is assumed that $y$ is independent of $x$.

In the partial derivative $\frac{\partial f(x,y)}{\partial x}$ $y$ is kept constant.
The total derivative $\frac{\text{d}f(x,y(x))}{\text{d}x}$ does NOT assume that $y$ is constant. The change of $x$ also affects $y$.

Example

For example, consider a function $f(x,y) = 3x^2 ~+~ 2y $ that depends on $x$ and $y$. A partial derivative of this function with respect to $ x $ is:

$$ \begin{align} \frac{\partial f(x,y)}{\partial x} ~&=~ \frac{\partial}{\partial x}(3x^2) ~+~ \frac{\partial}{\partial x}(2y) \\\\
~&=~ 6x \end{align} $$

The derivative of $2y$ with respect to $x$ is zero because $y$ is independent of $x$. A total derivative of the function, on the other hand, is:

$$ \begin{align} \frac{\text{d}f(x,y)}{\text{d}x} ~&=~ \frac{\text{d}}{\text{d} x}(3x^2) ~+~ \frac{\text{d}}{\text{d} x}(2y) \\\\
~&=~ 6x ~+~ 2 \frac{\text{d}y}{\text{d}x} \end{align} $$

Here the second term is generally not zero, because $y(x) $ can be a function of $ x $. If $y$ is independent of $x$, then the total derivative coincides with the partial derivative. Thus, a partial derivative is a special case of the total derivative.