how do you find the total derivative of a function that has 3 variables?

Question

for example, z = 4x^3 - 6xy - x^2 + 3y^2 + 12
is there any website that i can go to for help on calculus methods like this?

Answer 1

dz/dx = 12x^2 - (6y + 6x(dy/dx)) + 6y(dy/dx)

Try searching for implict differenciation on wikipedia...

Answer 2

If the function is differentiable, then the matrix representing the total derivative will just be the Jacobian- the matrix of partial derivatives.

In your example I think you're making z a function of the two variables x and y, so in fact you only have a function of two variables. In this case the Jacobian is the same as the gradient:

(dz/dx,dz/dy) = (12 x^2 -6y -2x, -6x + 6y)

Edit: Hmm, now I see that the term total derivative is used in several different ways. See wikipedia. I don't know which one you mean.

Answer 3

Right here. The total derivative of z be the sum of the derivative of z with respect to x and y. In each derivative, the other variable is treated as a constant.

So you will get 12x^2 -6y - 2x + 6y +12 for the x-partial or 12x^2-2x+12.
And for the y-partial -6x+6y.
Now add them up.

Answer 4

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