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 the point (0, e), we first need to find the derivative of the curve with respect to x.
The given equation is: x + (i * ny)^2 - e^xy = 0
To find the derivative, we need to differentiate each term with respect to x. Let's go through each term one by one:
1. Differentiating x with respect to x gives us 1.
2. To differentiate (i * ny)^2, we treat (i * ny) as a single constant. Differentiating a constant squared is as simple as multiplying it by 2, so we get 2(i * ny).
3. To differentiate e^xy, we use the chain rule. The derivative of e^u with respect to x is e^u * du/dx, where u = xy. Thus, the derivative is e^u * y.
So, differentiating the equation x + (i * ny)^2 - e^xy = 0 with respect to x gives us:
1 + 2(i * ny) - e^xy * y = 0
Now, to find the slope of the tangent at the point (0, e), we substitute x = 0 and y = e into the derivative:
1 + 2(i * e * e) - e^0 * e = 1 + 2ie^2 - e * e = 1 + 2ie^2 - e^2
Thus, the slope of the tangent to the curve at the point (0, e) is 1 + 2ie^2 - e^2.