Andre W. answered • 10/09/13

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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}=4is

2x e

^{y}+ x² y' e^{y}+y' e^{x}+ y e^{x}=0Evaluate this at (2,0) and get

4+4y'+y' e

^{2}=0so 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)

Leigh A.

10/09/13