Using the chain rule with 2 variables ?

[Rizzamabob]

Hey, I am a bit confused oh how to use the chain rule when i have 2 variables in an equation...
Example : f(x,y) = (Squareroot(x)).(cosh(x+y^2)) x(s,t)=st y(s,t)=s/t
When i have 2 variables, I am not sure how to split it up and use the chain rule, all the examples i found only have 1 variable :shy: Thanks :(

 

[benorin]

When differentiating a multivariate function with respect to one variable, you hold the other variables constant, e.g.
$$\frac{d}{dy}\left(x\sin(xy^{2})\right)=x\frac{d}{dy}\sin(xy^{2})=x\cos(xy^{2})\frac{d}{dy}(xy^{2})=2x^{2}y\cos(xy^{2})$$

in the above example x is a constant. Whereas in the example below, y is a constant:

$$\frac{d}{dx}\left(x\sin(xy^{2})\right)=\frac{d}{dx}\left(x\right)\cdot\sin(xy^{2})+x\frac{d}{dx}\left(\sin(xy^{2})\right)=1\cdot\sin(xy^{2})+x\cos(xy^{2})\frac{d}{dx}(xy^{2})=\sin(xy^{2})+xy^{2}\cos(xy^{2})$$

 

[Rizzamabob]

Ok, so say when i derive cosh(x + y^2) with respect to x, i get (1+y^2)sinh(x+y^2) correct ?? basically i don't touch y^2 and treat it as a whole number ?? :shy: But i don't get it, since if i had like cosh(x+3), then the derivative would just be sinh(x+3) using the chain rule correct ?? so why is the y^2 out the front in the other one ?

 

[Galileo]

When you differentiate with respect to one variable, say x, you treat the other, y, as constant. Specifically:

$$\frac{\partial}{\partial x}f(x,y)=\lim_{h\to 0}\frac{f(x +h,y)-f(x,y)}{h}$$
So it's just the rate of change of the function as you vary x you're interested in.
By the way, this alone has nothing to do with the chain rule, which tells you how to differentiate composite functions.

Now, differentiating cosh(x+y^2) with respect to x just means treating y constant. (So for this purpose we can treat the function as just dependent on x alone):
$$\frac{d}{dx}\cosh(x+y^2)=\sinh(x+y^2)\frac{d}{dx}(x+y^2)=\sinh(x+y^2)$$
Since the derivative of (x+y^2) wrt x is 1.