10
$\begingroup$

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

  • 5
    That's the [differential of the function](http://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables).2012-01-27

1 Answers 1

5

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,

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

  • 0
    @cohen Fixed. Thanks!2012-01-28