if e^xy = tan(y), then dy/dx=?

Answer 1

Use the implicit differentiation method to find dy/dx

since e^xy=tan(y), take the derivative for both sides, then

(e^xy)'=(tan(y))'

e^xyd(xy)/dx=sec²y dy/dx

e^xy(xdy/dx+y)=sec²y dy/dx (use product rule for the right hand side)

xe^xydy/dx+ye^xy=sec²y dy/dx

Now collect the dy/dx terms and find it. That is

sec²y dy/dx-xe^xydy/dx=ye^xy

dy/dx(sec²y-xe^xy)=ye^xy (factor the right side)

dy/dx=(ye^xy)/(sec²y-xe^xy)