Partial derivatives of multivariable functions f, g, h

[pointassist30]

Homework Statement

calculate the derivative of the following functions?

f(x,y,z,t) = (x-1)(2-y)z + (t^3 - 1)xyz
g(x,y) = 1/(1 + exp(-(ax + by + c))
h(x,y,z) = (x-1)^2 exp(x) + (y-2)^3 * z^3

The Attempt at a Solution

the way i was thinking was may be split the problem into multiple parts according to different variales. so if i have x,y,z in my problem...split into three parts and take derivative of one variable at a time. when taking a derivate, assume the other variables are constant.

not really sure how to do it though.

any help would be appreciated.

thanks.

 

[zhermes]

so if i have x,y,z in my problem...split into three parts and take derivative of one variable at a time. when taking a derivate, assume the other variables are constant

Thats exactly right, i.e.
$$df(x,y) = \frac{\partial f(x,y)}{\partial x} dx + \frac{\partial f(x,y)}{\partial y} dy$$

So, if you have something like $$f(x,y) = x^2y + x + y$$ you get:
$$df(x,y) = (2xy + 1)dx + (x^2 + 1)dy$$

When you take the derivative with respect to each variable, you pretend the other variables are all constant

 

[HallsofIvy]

Homework Statement

calculate the derivative of the following functions?

What do you mean by "the derivative" of a function of several variables? The partial derivatives or, as zhermes interpreted it, the differential?

f(x,y,z,t) = (x-1)(2-y)z + (t^3 - 1)xyz
g(x,y) = 1/(1 + exp(-(ax + by + c))
h(x,y,z) = (x-1)^2 exp(x) + (y-2)^3 * z^3

The Attempt at a Solution

the way i was thinking was may be split the problem into multiple parts according to different variales. so if i have x,y,z in my problem...split into three parts and take derivative of one variable at a time. when taking a derivate, assume the other variables are constant.

not really sure how to do it though.

any help would be appreciated.

thanks.