use the fact that the derivative of the function f(x)=3/x is f(x)=-3/x^2 to find the equation of the tangent…

Explanation:

Step1: Find the slope of the tangent line

The slope $m$ of the tangent line is the value of the derivative at the given point. Substitute $x = - 5$ into $f'(x)=\frac{-3}{x^{2}}$. $m=f'(-5)=\frac{-3}{(-5)^{2}}=-\frac{3}{25}$

Step2: Find the y - coordinate of the point on the curve

Substitute $x=-5$ into $f(x)=\frac{3}{x}$. $f(-5)=\frac{3}{-5}=-\frac{3}{5}$

Step3: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(-5,-\frac{3}{5})$ and $m =-\frac{3}{25}$. $y+\frac{3}{5}=-\frac{3}{25}(x + 5)$

Step4: Simplify the equation

First, distribute on the right - hand side: $y+\frac{3}{5}=-\frac{3}{25}x-\frac{3}{5}$ Then subtract $\frac{3}{5}$ from both sides: $y=-\frac{3}{25}x-\frac{6}{5}$

Answer:

$y =-\frac{3}{25}x-\frac{6}{5}$