determine the slope of the tangent to the curve x+(iny)^2 - e^xy = 0 at the point (0,e)
To determine the slope of the tangent to the curve at a given point, we can use the concept of differentiation.
Firstly, let's find the derivative of the equation with respect to x implicitly. We differentiate each term separately and apply the chain rule when needed.
Differentiating the equation x + (in^2)y^2 - e^xy = 0, we get:
1 + 2(in^2)y(dy/dx) - (y)(e^xy + x(dy/dx)) = 0
Next, let's substitute the given point (0, e) into this equation.
For the slope of the tangent at the point (0, e), we need to find the value of dy/dx when x = 0 and y = e.
Substituting x = 0 and y = e into the above equation, we have:
1 + 2(en^2)e(dy/dx) - (e)(1) = 0
Simplifying this equation, we get:
1 + 2(en^2)e(dy/dx) - e = 0
Now, let's solve for dy/dx, which gives us the slope of the tangent at the point (0, e).
Rearranging the equation, we have:
2(en^2)e(dy/dx) = e - 1
Dividing both sides by 2(en^2)e, we get:
dy/dx = (e - 1) / (2(en^2)e)
So, the slope of the tangent to the curve at the point (0, e) is given by (e - 1) / (2(en^2)e).