Let f(x,y,z) = g(x,z) Find all partial derivatives.

(df)/(dx) = (df)/(dy) = (df)/(dz) = g'(x,z)

Is the above true? If true...

(The actual homework question)

Let p(x,y,z) = q( q( x^2, xy), q( xyz, sin(x^2 * y^2 * z^3))) where q is a function of 2-variables. Find all partial derivatives.

What I have tried: By the chain rule, I only need to differentiate q(x^2, xy) and x^2 when differentiating w.r.t to x. (Apply a similar argument for ... w.r.t y)
and when I diff. w.r.t z, (dp)/(dz) = q'

Confusing enough? Opinions or facts/links are appreciated.

No. In this first part, the question is to understand what happens when a function of 3 variables (x,y,z) actuall depends only on 2 (x,z).

Since f doesn't depend on y, you can write : df/dy = 0.

As the 2 other variables are concerned :
df/dx = dg/dx
df/dz = dg/dz

Be very careful with the notation g' when g is a function of several variables. You shouldn't use it... at all.



dp/dx = dq/dx ( q(x²;xy) ; q( xyz ; sin A)) . ( dq/dx ( x² ; xy).2x + dq/dy (x² ; xy).y) + dq/dy ( q(x²;xy) ; q( xyz ; sin A)) . ( dq/dx ( xyz ; A ).yz + dq/dy ( xyz ; A ). 2x y² z^3. cos (x²y²z^3) )

where A = sin (x²y²z^3)

If you get lost somewhere, I strongly advice you to try with something much less complicated as a training.