When working through a problem set containing Implicit Differentiation problems, I've found that I keep getting the wrong answer compared to the one listed at the back of my book.
The problem is given as such: Use implicit differentiation to find the equation of the tangent line to the curve at a given point
x^2 + xy + y^2 = 3
With given point (1, 1). I also am told that it is an ellipse.
To solve this, I evidently must differentiate both sides of the problem:
1: dy/dx ( x^2 + xy + Y^2 ) = dy/dx(3)
2: dy/dx (2x + 1y'+ 2yy') = 0
3: 1y' + 2yy' = 0 - 2x
4: y'(1+2y) = -2x
5: y' = -2x/(1+2y)
Hurray, so now since I have the first derivative of Y. I can use it to find the slope at the point.
Slope at Point (1,1)= -2(1) / (1+2(1)
Slope at Point (1,1)= -2/3
So now that I've got my slope, I know the equation of the tangent will be in the form:
So, given I now know the slope:
y=-2/3x + b
Substitute in the known point:
1 = -2/3(1) + b
b = 5/3
So the final answer I get is: y = -2/3x + 5/3
But according to the answer, it is supposed to be: -x + 2, I don't know where I went wrong, and I've done it twice to make sure I'm getting the same answer. Could someone please help me?