# Math Help - Differentation Question

1. ## Differentation Question

Hi everyone

Need help on this question, can someone help me.Really appreciate all help & support.

Find $\frac {dz}{dx}$ & $\frac{dz}{dy}$for
$xyz-3yz^2-4xy^2=0$

Thank you in advance for all help & support.

2. Just in case a picture helps...

... where

... is the product rule, and...

... the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Follow through by solving the bottom row for $\frac{\partial z}{\partial x}$ , of course.

w.r.t. y ...

Spoiler:

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

3. thank you for replying. really hope someone can help with the working then i'll understand better.

thank you for all help & support,really appreciate.

4. Originally Posted by anderson
Hi everyone

Need help on this question, can someone help me.Really appreciate all help & support.

Find $\frac {dz}{dx}$ & $\frac{dz}{dy}$for
$xyz-3yz^2-4xy^2=0$

Thank you in advance for all help & support.
Differentiate both sides of the equation with respect to x
(treating z as a function of x and y as a constant):

$(xyz)_x- (3yz^2)_x- (4xy^2)_x= 0$
$= yz+ xyz_x- 6yz z_x- 4y^2= 0$

Now solve for $z_x$

Differentiat both sides of the equation with respect to y
(treating z as a function of y and x as a constant):
$(xyz)_y- (3yz^2)_y- (4xy^2)_y= 0$

You finish that.

5. thank you for explaining, just wondering how we get the $xyz_x$.

thank you so much for all help & support.

6. hi everyone

thank you for guiding. is this correct, please help confirm.thank you for helping me to learn, really appreciate all help & support.

$

xyz-3yz^2-4xy^2=0
$

$yz+xy\frac{dz}{dx}-6yz\frac{dz}{dx}-4y^2$
$\frac{dz}{dx}(xy-6yz)=4y^2-yz$
$\frac{dz}{dx}=\frac{4y^2-yz}{xy-6yz}$

$xz+xy\frac{dz}{dy}-3z^2-6yz\frac{dz}{dy}-8xy=0$
$\frac{dz}{dy}(xy-6yz)=3z^2+8xy-xz$
$\frac{dz}{dy}=\frac{3z^2+8xy-xz}{xy-6yz}$

7. Correct