1. ## Partial derivative, multivariable

$f(x,y)$

For all s,t :
$f(s,8s)=6s+cos(6s)$
$f(t,-7t)=1+6t+5t^2$

Determin $\frac{\partial f}{\partial x}(0,0)$ and $\frac{\partial f}{\partial y}(0,0)$

Im new to multi variables so i donīt know how to do this.

I thought if i set $z=f(x(s,t),y(s,t))$

i get:

$\frac{\partial z}{\partial s }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$

$\frac{\partial z}{\partial t }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$

from here i donīt know what to do.

Thanks!

2. Originally Posted by mechaniac
$f(x,y)$

For all s,t :
$f(s,8s)=6s+cos(6s)$
$f(t,-7t)=1+6t+5t^2$

Determin $\frac{\partial f}{\partial x}(0,0)$ and $\frac{\partial f}{\partial y}(0,0)$

Im new to multi variables so i donīt know how to do this.

I thought if i set $z=f(x(s,t),y(s,t))$

i get:

$\frac{\partial z}{\partial s }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$

$\frac{\partial z}{\partial t }= \frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$

from here i donīt know what to do.

Thanks!
You have a good start. Using the chain rule that you have above gives

$\frac{\partial f}{\partial s}=6-6\sin(6s)=\frac{\partial f}{\partial x}\cdot 1+\frac{\partial f}{\partial y}\cdot 8$

Using the subscript notation for partial derivatives gives
$f_x(s,8s)+8f_y(s,8s)=6-6\sin(6s)$

$\frac{\partial f}{\partial t}=6+10t=\frac{\partial f}{\partial x}\cdot 1+\frac{\partial f}{\partial y}\cdot (-7)$

$f_x(t,-7t)-7f_y(t,-7t)=6+10t$

Now if
$t=s=0$

You will have a system of equations to find the partial derivatives you want!

3. Thanks! i just got confused by all the variables