Question

question 5
the slope of the tangent line to the curve $y=\frac{2}{x}$ at the point $\left(9,\frac{2}{9}\
ight)$ is:

the equation of this tangent line can be written in the form $y=mx+b$ where:
m is:
b is:

Explanation:

Step1: Rewrite the curve function

$y = 2x^{-1}$

Step2: Find derivative for slope

$\frac{dy}{dx} = -2x^{-2} = -\frac{2}{x^2}$

Step3: Calculate slope at $x=9$

$m = -\frac{2}{9^2} = -\frac{2}{81}$

Step4: Solve for $b$ using point $(9,\frac{2}{9})$

$\frac{2}{9} = -\frac{2}{81}(9) + b$
$\frac{2}{9} = -\frac{2}{9} + b$
$b = \frac{2}{9} + \frac{2}{9} = \frac{4}{9}$

Answer:

Slope: $\boldsymbol{-\frac{2}{81}}$
$m$: $\boldsymbol{-\frac{2}{81}}$
$b$: $\boldsymbol{\frac{4}{9}}$
Equation: $\boldsymbol{y = -\frac{2}{81}x + \frac{4}{9}}$