Question

consider the following.\\( y = \ln(x^9),\\ (1, 0)\\)\\(\text{(a)}\\) find \\(y\\).\\(y = \\)\\(\text{find the slope of the line tangent to the graph of } y \text{ at the given point.}\\)\\(\text{find an equation of the tangent line to the graph of the function at the given point.}\\)\\(y = \\)\\(\text{(b)}\\) use a graphing utility to graph the function and its tangent line at the point. use the tangent featu plotted on the screen. select the submit button to grade your response.)

Explanation:

Step1: Simplify the original function

Use logarithm power rule: $\ln(x^n)=n\ln(x)$.
$y = 9\ln(x)$

Step2: Differentiate the simplified function

Apply derivative rule: $\frac{d}{dx}[\ln(x)]=\frac{1}{x}$.
$y' = 9 \cdot \frac{1}{x} = \frac{9}{x}$

Step3: Find tangent slope at $(1,0)$

Substitute $x=1$ into $y'$.
$m = \frac{9}{1} = 9$

Step4: Derive tangent line equation

Use point-slope form: $y - y_1 = m(x - x_1)$, where $(x_1,y_1)=(1,0)$ and $m=9$.
$y - 0 = 9(x - 1)$
Simplify to slope-intercept form: $y = 9x - 9$

Answer:

(a)
$y' = \frac{9}{x}$
Slope of tangent line: $9$
Equation of tangent line: $y = 9x - 9$
(b)
When graphing, plot the function $y=\ln(x^9)$ (equivalent to $y=9\ln(x)$ for $x>0$) and the line $y=9x-9$. The tangent line will touch the function only at the point $(1,0)$.