# differentiation question

• December 29th 2008, 05:15 AM
oligo
differentiation question
Could somebody help me 'total differentiate' this question please?

K = 12x^3y + 10x^2y^2 + 6xy^3 +4y^4

I don't really know where to begin this is what i have down

fx = 36x^2 y + 20xy^2 + 6y^3 +4y^4
fy = 12x^3 + 10x^2(2y) + 6x(3y^2) + 16y^3

then im lost on the total differential formula
DK = dk/dx (DK) + dk/dy (DY)

(where the small 'd' represents the differential symbol

am i completely lost?
• December 29th 2008, 07:39 AM
Jester
Quote:

Originally Posted by oligo
Could somebody help me 'total differentiate' this question please?

K = 12x^3y + 10x^2y^2 + 6xy^3 +4y^4

I don't really know where to begin this is what i have down

fx = 36x^2 y + 20xy^2 + 6y^3 +4y^4
fy = 12x^3 + 10x^2(2y) + 6x(3y^2) + 16y^3

then im lost on the total differential formula
DK = dk/dx (DK) + dk/dy (DY)

(where the small 'd' represents the differential symbol

am i completely lost?

The total differential is gven by

$dK = \frac{\partial K}{\partial x} \,dx +\frac{\partial K}{\partial y} \,dy$
where

$
K_x = 36x^2 y + 20xy^2 + 6y^3$

$
K_y = 12x^3 + 10x^2(2y) + 6x(3y^2) + 16y^3
$

(note the differences in our derivatives)
• December 29th 2008, 07:42 AM
HallsofIvy
If F(x,y) is a function of x and y, and x and y are themselves functions of some variable t, then we could write F(t)= F(x(t),y(t)) as a function of t and, by the chain rule, $\frac{dF}{dt}= \frac{\partial F}{\partial x}\frac{dx}{dt}+ \frac{\partial F}{\partial y}\frac{dy}{dt}$. Since the differential of a function F(t) is defined as [tex]dF= \frac{dF}{dt}dt[tex], that gives, as the "total differential" $dF= \frac{\partial F}{\partial x}dx+ \frac{\partial F}{\partial y}dy$

You almost have the correct partial derivatives
$Kx = 36x^2 y + 20xy^2 + 6y^3 +4y^4$
is not quite right. The derivative of $4y^4$ with respect to x is 0, not $4y^4$.

$Ky = 12x^3 + 10x^2(2y) + 6x(3y^2) + 16y^3$
is correct- once you have multiplied 10(2)= 20 and 6(3)= 18.

Then, to find dK by simply writing
$dK= (36x^2 y + 20xy^2 + 6y^3)dx+ (12x^3 + 20x^2y + 18xy^2 + 16y^3)dy$