# How do you implicitly differentiate #y= e^(xy)-xy^2 #?

##### 1 Answer

#### Explanation:

The derivative of the whole function is

#d/dx(y)=d/dx(e^(xy))-d/dx(xy^2)#

This will be more manageable if we split it up into each individual part first.

#d/dx(y)=ul(color(blue)(dy/dx#

The next is a little trickier. We should recognize that chain rule will be necessary, since

#d/dx(e^(xy))=d/dx(xy)*e^(xy)#

To find

#d/dx(xy)=yd/dx(x)+xd/dx(y)=y+xdy/dx#

Plug this back in to the previous expression to see that

#d/dx(e^(xy))=ul(color(blue)((y+xdy/dx)e^(xy)#

For this last part, use the product rule once more.

#d/dx(-xy^2)=-y^2d/dx(x)-xd/dx(y^2)#

To find the derivative of

#d/dx(y^2)=2ydy/dx#

Which gives the derivative of the final part:

#d/dx(-xy^2)=ul(color(blue)(-y^2-2xydy/dx#

Plug these all back in for the derivative of the entire equation:

#dy/dx=(y+xdy/dx)e^(xy)-y^2-2xydy/dx#

Now, solve for

#dy/dx-xe^(xy)dy/dx+2xydy/dx=ye^(xy)-y^2#

#dy/dx(2xy-xe^(xy)+1)=ye^(xy)-y^2#

#dy/dx=(ye^(xy)-y^2)/(2xy-xe^(xy)+1)#