Question

consider the equation and the given point.\\( f(x) = \frac{x}{x + 6}, \\ (-7, 7) \\)\\( \text{(a) find an equation of the tangent line to the graph of } f \text{ at the given point.} \\)\\( y = \frac{8}{9}x + \frac{85}{9} \\) (marked incorrect)\\( \text{(b) use a graphing utility to graph the function and its tangent line at the point. use the tangent feature...} \\)

Explanation:

Step1: Find derivative of $f(x)$

Use quotient rule: if $f(x)=\frac{u}{v}$, $f'(x)=\frac{u'v-uv'}{v^2}$.
Here $u=x$, $u'=1$; $v=x+6$, $v'=1$.

$$ f'(x)=\frac{1\cdot(x+6)-x\cdot1}{(x+6)^2}=\frac{x+6-x}{(x+6)^2}=\frac{6}{(x+6)^2} $$

Step2: Compute slope at $x=-7$

Substitute $x=-7$ into $f'(x)$:

$$ f'(-7)=\frac{6}{(-7+6)^2}=\frac{6}{(-1)^2}=6 $$

Step3: Use point-slope form

Point-slope formula: $y-y_1=m(x-x_1)$, where $(x_1,y_1)=(-7,7)$, $m=6$.

$$ y-7=6(x-(-7)) $$

Step4: Simplify to slope-intercept form

Expand and isolate $y$:

$$ y-7=6x+42 \implies y=6x+49 $$

Answer:

$y=6x+49$