Find an equation of the tangent line to the curve (x^2)(e^y)+(ye^x)=4 at the point (2,0).

Question

I think you're supposed to use implicit differentiation, and then solve for the equation after finding the slope.

Answer 1

You can either use implicit differentiation or the implicit function theorem to find y'(2). The implicit (or total) derivative of

 

x² e^y + y e^x =4

 

is

 

2x e^y + x² y' e^y +y' e^x + y e^x =0

 

Evaluate this at (2,0) and get

 

4+4y'+y' e²=0

 

so that

 

y'(2)= -4/(4+e²)

 

This is the slope of the tangent line to the implicit curve at (2,0). To find its equation plug in this slope and the point (2,0) into the standard linear equation

 

y=mx+b

 

and get

 

0=(-4/(4+e²)) 2 +b

 

b=8/(4+e²)

 

so the tangent line equation is

 

y=-4/(4+e²) x + 8/(4+e²) = 4/(4+e²) (-x+2)