Question

consider the following. (the graph is called a witch of agnesi.)
find the derivative of the function.
f(x) = $\frac{-18x}{(x^2 + 5)^2}$
find f(x) when x = 2.
f(2) = -4/9
give the slope of the graph of the function at the point (2, 1).
slope = -4/9
find an equation of the tangent line to the graph of f at the point (2, 1).
y = $\frac{7}{162}x + \frac{1183}{162}$

Explanation:

Step1: Recall point-slope formula

The point-slope form of a line is $y - y_1 = m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.
We know $x_1=2$, $y_1=1$, and $m=f'(2)=-\frac{4}{9}$.

Step2: Substitute values into formula

$y - 1 = -\frac{4}{9}(x - 2)$

Step3: Simplify to slope-intercept form

First expand the right-hand side:
$y - 1 = -\frac{4}{9}x + \frac{8}{9}$
Then add 1 (which is $\frac{9}{9}$) to both sides:
$y = -\frac{4}{9}x + \frac{8}{9} + \frac{9}{9}$
$y = -\frac{4}{9}x + \frac{17}{9}$

Answer:

$y = -\frac{4}{9}x + \frac{17}{9}$