Question 9 Find the equation of the tangent line to the curve \( f(x)=\frac{1}{2 x+1} \) at \( x=0 \). Your equation must be in the form \( y=m x+b \).

To find the equation of the tangent line, we first need to determine the derivative \( f'(x) \) of the function \( f(x) = \frac{1}{2x + 1} \). Using the quotient rule, we can calculate that \( f'(x) = -\frac{2}{(2x + 1)^2} \). Next, evaluate \( f(0) \) to find the y-coordinate of the point of tangency: \[ f(0) = \frac{1}{2(0) + 1} = 1. \] Now, substitute \( x = 0 \) into the derivative to find the slope of the tangent line: \[ f'(0) = -\frac{2}{(2(0) + 1)^2} = -2. \] With the slope \( m = -2 \) and the point (0, 1), we can use the point-slope form to find the equation of the tangent line: \[ y - 1 = -2(x - 0), \] which simplifies to \[ y = -2x + 1. \] So, the equation of the tangent line is: \[ y = -2x + 1. \]