Final answer:
The equation of the tangent line to y = 1/(x^2 + 1) at x = 2 is y = -4/25x + 11/25.
Step-by-step explanation:
To find the equation of the tangent line to the curve y = 1/(x^2 + 1) at x = 2, we need to find the derivative of the curve and evaluate it at x = 2. The derivative of y = 1/(x^2 + 1) is y' = -2x/(x^2 + 1)^2.
Substituting x = 2 into the derivative gives y'(2) = -2(2)/(2^2 + 1)^2 = -4/25.
Using the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, we can substitute x = 2, y = 1/(2^2 + 1) = 1/5, and m = -4/25 to find the equation of the tangent line:
y - 1/5 = -4/25(x - 2)
Simplifying the equation gives:
y = -4/25x + 11/25