Provide an appropriate response.
-$$\text { Find an equation for the tangent to the curve } y = \frac { 8 x } { x ^ { 2 } + 1 } \text { at the point } ( 1,4 ) \text {. }$$
A) y = 4
B) y = x + 4
C) y = 0
D) y = 4x

Question

Quizplus · Accepted Answer

The slope of the tangent line at a point on a curve is given by the derivative of the curve at that point. For the given curve $y = \frac{8x}{x^2 + 1}$, we first find its derivative using the quotient rule: $$y' = \frac{(x^2 + 1)(8) - 8x(2x)}{(x^2 + 1)^2} = \frac{8 - 8x^2}{(x^2 + 1)^2}.$$ At $x = 1$, the slope of the tangent line is $$y'(1) = \frac{8 - 8(1)^2}{(1^2 + 1)^2} = \frac{0}{4} = 0.$$ This means the tangent line is horizontal. Since it passes through (1,4), its equation is $y = 4$.