It is proved that the area of the triangle formed by the tangent line at a point on the curve f (x) = 1 / x, the line x = 0 and the line y = x is a fixed value, and And calculate the fixed value

It is proved that the area of the triangle formed by the tangent line at a point on the curve f (x) = 1 / x, the line x = 0 and the line y = x is a fixed value, and And calculate the fixed value

Take the first quadrant as an example
Let the tangent point be (x1,1 / x1) and use the derivative to know the square of the tangent slope of - 1 / x1. If you don't know the derivative, use the most stupid method to find the tangent. Therefore, the tangent passing through the tangent point is Y-1 / X1 = - 1 / x1 square * (x-x1), so the ordinate of the intersection of the tangent and the straight line x = 0 is 2 / x1
So the triangle area is 2 / X1 * X1 * 1 / 2 = 1