To find the equation of the tangent line to the curve sin(x+y) = 2x - 2y at the point (π, π), we first need to find the derivative of the curve with respect to x.
Implicit differentiation:
Differentiating both sides with respect to x using chain rule, we have:
cos(x+y)(1+dy/dx) = 2 - 2(dy/dx) ⟹ cos(x+y)(1+dy/dx) - 2 + 2(dy/dx) = 0
Now we need to solve for dy/dx:
(dy/dx)(2+cos(x+y)) = 2 - cos(x+y) ⟹ dy/dx = (2 - cos(x+y)) / (2 + cos(x+y))
Now we calculate dy/dx at the point (π, π):
dy/dx = (2 - cos(2π)) / (2 + cos(2π)) = (2 - 1) / (2+1) = 1/3.
Now we have the slope (dy/dx) of the tangent line at the point (π, π).
To find the equation of the tangent line, we use the point-slope form:
y - y1 = m(x - x1)
Where m = dy/dx, x1 = π and y1 = π. Plugging in the values, we get:
y - π = 1/3 (x - π)
Now, we can rewrite the equation of the tangent line to a more standard form:
y = (1/3)x - (1/3)π + π
So, the equation of the tangent line is:
y = (1/3)x + (2/3)π.
Use implicit differentiation to find an equation of the tangent line to the curve
sin(x+y)=2x−2y at the point (π,π).
Tangent Line Equation:
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