a. find the derivative function f for the function f. b. determine an equation of the line tangent to the…

Answer:

b. $y = - 3x - 6$

Explanation:

Step1: Find $f(a)$

Given $f(x)=\frac{6}{4x + 1}$ and $a=-1$. Then $f(-1)=\frac{6}{4\times(-1)+1}=\frac{6}{-3}=- 2$.

Step2: Find the slope of the tangent - line

The derivative $f^{\prime}(x)=-\frac{24}{(4x + 1)^{2}}$. When $x = a=-1$, the slope $m=f^{\prime}(-1)=-\frac{24}{[4\times(-1)+1]^{2}}=-\frac{24}{(-3)^{2}}=- \frac{24}{9}=-3$.

Step3: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(a,f(a))=(-1,-2)$ and $m = - 3$. Substitute into the formula: $y-(-2)=-3(x - (-1))$. Simplify: $y + 2=-3(x + 1)$. Expand: $y+2=-3x - 3$. Solve for $y$: $y=-3x - 6$.