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)#
