Respuesta :
Answer:
The equation of the tangent line to the curve
3 x - y = 2
Step-by-step explanation:
Step(i):-
Given function = f(x,y) = [tex]2e^{xy} -x-y=0[/tex] ...(i)
Differentiating equation (i) with respective to 'x' , we get
[tex]2 e^{xy} \frac{d}{d x} (x y) -1 -\frac{dy}{dx} =0[/tex]
apply formula
[tex]\frac{d}{dx} (UV) = UV^{l} +V U^{l}[/tex]
step(ii):-
⇒ [tex]2 e^{xy} (x(\frac{d}{d x} ( y))+y(1)) -1 -\frac{dy}{dx} =0[/tex]
⇒ [tex]2 e^{xy} (x(\frac{d}{d x} ( y))+ 2e^{xy} y(1)) -1 -\frac{dy}{dx} =0[/tex]
Taking common d y/d x
[tex](2 e^{xy} (x) -1)\frac{dy}{dx} =1- 2e^{x y} y(1))[/tex]
[tex]\frac{dy}{dx} =\frac{1- 2e^{x y} y(1))}{(2 e^{xy} (x) -1)}[/tex]
put At (0,2)
[tex]\frac{dy}{dx} =\frac{1- 2e^{0} 2(1))}{(2 e^{0} (0) -1)}=\frac{1-4}{-1} =3[/tex]
slope of the curve m = 3
Step(iii):-
The equation of the tangent line to the curve
[tex]y-y_{1} =m(x-x_{1} )[/tex]
y - 2 = 3 ( x - 0 )
3 x - y = 2
Final answer:-
The equation of the tangent line to the curve
3 x - y = 2
