Partial Derivative of Composite Functions

[darkp0tat0]

Any help would be much appreciated - Is it possible to say the following?

If z = g(s+at) + f(s-at), let u = s+at and v=s-at, where a is a constant.

z = g(u) + f(v), $\frac{∂z}{∂u}$ = g'(u), $\frac{∂^{2}z}{∂v∂u}$ = 0?

or can ∂u and ∂v not even exist because it depends on two variables (a and t), which are the same ones as the ones v depends on.

Thanks!

 

[HallsofIvy]

"$$\partial u$$" does not exist by itself. While we define "differentials", dx and dy, we do NOT define "partial differentials". Rather, if f(x, y) is a function of the two variables, x and y, we define its "total differential":
$$df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}$$

If If z = g(s+at) + f(s-at), let u = s+at and v=s-at, where a is a constant, and z = g(u) + f(v), then
$$\frac{\partial z}{\partial s}= \frac{\partial f}{\partial s}+ \frac{\partial g}{\partial s}$$
$$= \frac{df}{du}\frac{\partial u}{\partial s}+ \frac{dg}{dv}\frac{\partial v}{\partial s}= \frac{df}{du}(1)+ \frac{dg}{dv}(1)= \frac{df}{du}+ \frac{dg}{dv}$$

Similarly
$$\frac{\partial z}{\partial t}= a\frac{df}{du}- a\frac{dg}{dv}$$

 

[darkp0tat0]

Thanks for your quick reply! I think I get what you mean by ∂u can't exist by itself. Do you think you would be able to take a look at the original equation I posted:

$z = g(u) + f(v), \frac{∂^{2}z}{∂u∂v} = 0$

Does that still hold true even when you can't hold v or u constant while changing the other? (they depend on the same two vars)