Final answer:
The equation of the line tangent to the curve y = 4/(x²+1) at the point (1,2) is found by first calculating the derivative of y, then using the point-slope form to find the tangent line's equation, which simplifies to y = -2x + 4.
Step-by-step explanation:
To find the equation of the line tangent to y = 4/(x²+1) at the point (1,2), we must first compute the derivative of the function to find the slope of the tangent line. The derivative of y with respect to x is the slope of the tangent line at any point on the curve. To find this derivative, we use the quotient rule, which states that if y = u/v, then y' (the derivative of y) is (vu' - uv')/v², where u' and v' are the derivatives of u and v, respectively, with respect to x.
For the function y = 4/(x²+1), u = 4 and v = x²+1. Therefore, u' = 0 and v' = 2x. Substituting these into the quotient rule gives us the derivative y' = -(8x)/(x²+1)². Now we find the slope at x = 1 by substituting x into y', giving y'(1) = -8/(1+1)² = -2.
The slope of the tangent line at (1,2) is -2. The equation of the tangent line can be expressed in point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the tangent line and m is the slope. Using the point (1,2) and slope -2, we get y - 2 = -2(x - 1).
Simplify the equation to get y = -2x + 4, which is the equation of the line tangent to the curve at the point (1,2).