TITLE: Calculus - can I do this?
QUESTION [10 upvotes]: 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

REPLY [5 votes]: 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, 
$$\mathrm{d}y=\sum_{i=1}^n \dfrac{\partial y}{\partial x_i}\cdot\mathrm{d}x_i$$