# Partial Differentiation

• May 28th 2010, 08:47 AM
wkn0524
Partial Differentiation
Find : ∂z/∂x
$\displaystyle z = 3x\sqrt{y} - xcos(xy)$

My solution:
$\displaystyle Let K = xcos(xy)$
$\displaystyle Let u = x , v = cos(xy)$
∂u/∂x = 1
∂v/∂x = $\displaystyle -ysin(xy)$
Product rule:
∂K/∂x = v(∂u/∂x) + u(∂v/∂x) = $\displaystyle cos(xy) - xysin(xy)$

∂z/∂x = $\displaystyle 3\sqrt{y} -cos(xy) -xysin(xy)$
• May 28th 2010, 08:55 AM
General
There is no need to make substitutions ..
• May 28th 2010, 08:59 AM
wkn0524
Sorry, im learning how to using Math equation...my final answer is ∂z/∂x =$\displaystyle 3\sqrt{y} -cos(xy) -xysin(xy)$. Im just want to ask whether im correct or not.
• May 28th 2010, 09:05 AM
General
To write the square root of y : \sqrt{y}
You need the product rule to differentiate the second term,,
• May 28th 2010, 09:12 AM
wkn0524
Quote:

Originally Posted by General
To write the square root of y : \sqrt{y}
You need the product rule to differetiate the second term,,

Thanks. What you mean by "the product rule to differetiate the second term". (Worried)
• May 28th 2010, 09:15 AM
General
Quote:

Originally Posted by wkn0524
$\displaystyle z = 3x\sqrt{y} - { \color{red} xcos(xy) }$

How can you differentiate the red one ?
• May 28th 2010, 09:18 AM
wkn0524
Quote:

Originally Posted by General
How can you differentiate the red one ?

Sorry...The question is ask to find ∂z/∂x for this question $\displaystyle z = 3x\sqrt{y} - xcos(xy)$

After differentiate xcos(xy), then i get$\displaystyle cos(xy) - xysin(xy)$
• May 28th 2010, 09:19 AM
General
I know,
You have:
$\displaystyle z=3x\sqrt{y} - x \, cos(xy)$

And you want to find $\displaystyle z_x$ , Right ?
• May 28th 2010, 09:26 AM
wkn0524
Quote:

Originally Posted by General
I know,
You have:
$\displaystyle z=3x\sqrt{y} - x \, cos(xy)$

And you want to find $\displaystyle z_x$ , Right ?

Is ∂z/∂x =$\displaystyle z_x$ ?

If same, then yes.
• May 28th 2010, 09:31 AM
General
Yes.
They are the same.

$\displaystyle z=3x\sqrt{y} - x \, cos(xy)$

$\displaystyle z_x=3\sqrt{y} - \left[ { \color{blue} cos(xy) - xy sin(xy) } \right]$

I used the product in blue,,
• May 28th 2010, 09:36 AM
wkn0524
Quote:

Originally Posted by General
Yes.
They are the same.

$\displaystyle z=3x\sqrt{y} - x \, cos(xy)$

$\displaystyle z_x=3\sqrt{y} - \left[ { \color{blue} cos(xy) - xy sin(xy) } \right]$

I used the product in blue,,

Ouch, got the minus sign. The partial differentiation was not exactly same as differentiation topic, knew the concepts now. (Itwasntme)