a. For the following function, find \( f^{\prime}(a) \). b. Determine an equation of the line tangent to the graph of \( f \) at \( (a, f(a)) \) for the given value of a. \( f(x)=\frac{4}{x+3}, a=6 \)

To find \( f^{\prime}(a) \), we need to differentiate the function \( f(x) = \frac{4}{x+3} \). Using the power rule and the chain rule, we have: \[ f^{\prime}(x) = -\frac{4}{(x+3)^2} \] Now, substituting \( a = 6 \): \[ f^{\prime}(6) = -\frac{4}{(6+3)^2} = -\frac{4}{9^2} = -\frac{4}{81} \] For the tangent line equation at the point \( (6, f(6)) \), first we find \( f(6) \): \[ f(6) = \frac{4}{6+3} = \frac{4}{9} \] The equation of the tangent line can be expressed as: \[ y - f(6) = f^{\prime}(6)(x - 6) \] Substituting the known values: \[ y - \frac{4}{9} = -\frac{4}{81}(x - 6) \] Rearranging gives us the desired equation of the tangent line. Now go on and have fun exploring the results!