Notation Question - Expressing Rate of Change w/o Introducing Variable

[gz_student]

Hi All,

I have a question about notation.

Suppose I have an expression:

$$f(x,g(x,y))$$

I would like to know how to express (not calculate) the rate of change of the above expression with regards to x.

I can always express it like this:

Let $$z(x,y) = f(x,g(x,y))$$. Rate of change is $$\frac{{\partial z}}{{\partial x}}$$.

But that is awkward. Is there any way to express $$\frac{{\partial z}}{{\partial x}}$$ without having to introduce a variable z (i.e. using only variables and function names f,g,x,y)?

Thanks,
GZ

 

[ice109]

what's wrong with partial f/ partial x ?

ahh i see

what you want is

$$\frac{\partial f\big|_{y=g}}{\partial x}$$

or if you feel there might be ambiguity about whether the derivative is evaluated at g or f

$$\frac{\partial (f\big|_{y=g})}{\partial x}$$

honestly though

$$\frac{\partial }{\partial x} f(x,g)$$ is probably best

 

[gz_student]

Hi ice109,

Thank you for your response.

What I really want is the quantity $$\frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$.

$$\frac{\partial f}{\partial x}$$ does not reflect the second term above.

But do I have to introduce a new variable z in order to express this clearly? Or is there a better way?


I have never see the notation $$\frac{\partial}{\partial x} f(x,g)$$ before. Does it equal $$\frac{{\partial f}}{{\partial x}} +\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$?

Thanks,
gz