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

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1 Answer

You can either use implicit differentiation or the implicit function theorem to find y'(2). The implicit (or total) derivative of
 
x² ey + y ex =4
 
is
 
2x ey + x² y' ey +y' ex + y ex =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)
 

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