Apologies if this is a dumb question. I learned at school that I can differentiate $$y=x^{2}$$ to give $$\frac{dy}{dx}=2x.$$ But, if I have a multivariable function, for example$$y=4x^{2}+3z+t^{3}$$ am I allowed to differentiate it to give$$dy=8x\;dx+3\;dz+3t^{2}\;dt$$ and, if valid, what is this procedure called exactly?

Thank you


Yes it is valid and is called the differential of a function. In the link is the wikipedia page on this concept!

Consider the function, $$y=f(x_1,x_2,\cdots,x_n)$$

Goursat, a French Mathematician introduced the concept of partial differential of $y$, say, with respect to $x_i$.

A partial differential of $y$ with respect to $x_i$ is given by, $$\dfrac{\partial y}{\partial x_i}\cdot \mathrm{d}x_i$$

A total differential is the sum of the partial differentials of all the independent variables. So, it is the following,

$$\mathrm{d}y=\sum_{i=1}^n \dfrac{\partial y}{\partial x_i}\cdot\mathrm{d}x_i$$

  • $\begingroup$ Thanks - is that the same as the the "total differential"? It seems to be from the Wikipedia link. $\endgroup$ – Peter4075 Jan 27 '12 at 20:46
  • $\begingroup$ Yes, what you have shown us is in fact the total differential. I'll edit to add this! $\endgroup$ – user21436 Jan 27 '12 at 20:51
  • $\begingroup$ Also, you wrote "partial differential of y, say, with respect to $y_i$". I assume you meant $x_i$. $\endgroup$ – marty cohen Jan 28 '12 at 4:28
  • $\begingroup$ @cohen Fixed. Thanks! $\endgroup$ – user21436 Jan 28 '12 at 10:11

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