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I have to calculate the total derivative of the function

$$f(x,y) = \cos(x)\sin(y).$$

I found the total differential which is

$$d(f(x,y)) = -\sin(x)\sin(y)dx + \cos(x)\cos(y)dy.$$

Is my answer correct or there is a difference between total derivative/differential?

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  • $\begingroup$ You need to verify it against the definitions, which may vary between authors. For some authors the total derivative is only meaningful when $x$ and $y$ are themselves functions of a single variable $t.$ $\endgroup$ Commented Dec 6, 2015 at 15:25
  • $\begingroup$ The total derivative might mean $df$ or might mean the matrix $\pmatrix{\partial_x f & \partial_y f}$ or it might mean the linear transformation $\mathbf x \mapsto D\mathbf x$ where $D$ is the matrix defined above depending on how your professor defines it. $\endgroup$
    – user137731
    Commented Dec 6, 2015 at 15:29
  • $\begingroup$ after all your calculation is right $\endgroup$
    – janmarqz
    Commented Dec 6, 2015 at 16:38

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First, yes, your differential is correct. To see your function and its total differential see this page.

A differential is an expression for arbitrarily small changes (for one variable). Normally, the differentials are terms like $dx, dy, dz, ...$. The words differential and derivative are used in very similar or equal senses. But I would say, there is a difference of perspective.

A total derivative compares changes in two dimensions, so $\frac{df}{dx}$ means that we are interested in the change of $f$ (which is a function of $x$ and other variables!) in comparison to the changes in $x$.

To regard the changes in one variable independently from other variables (which $f$ depends on), the total variable considers the changes of other variables of $f$ in $x$ to (if you regard the total differential of $x$).

You can look up the formula here. As you can see, to form the total differential of $f$ (that is called $df$), you need the total differentials of all variables considered by $f$.

What you get is a differential - but now, it is in $df$. Before, you had changes compared between $f$ and $x$, $f$ and $y$ or $f$ and $z$. These are derivatives and so they create relations between changes. $df$ is no comparison, it is a total change, a differential.

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