To solve the problem and find the value of [tex]\(\frac{d y}{d x}\)[/tex] at the point [tex]\((1,1)\)[/tex] for the function [tex]\(y = \frac{(1+3 x)^7}{x^9}\)[/tex], we'll follow these steps:
1. Differentiate the function [tex]\(y\)[/tex] with respect to [tex]\(x\)[/tex] to find [tex]\(\frac{dy}{dx}\)[/tex]:
Given [tex]\(y = \frac{(1 + 3x)^7}{x^9}\)[/tex].
To differentiate this function, we can use both the quotient rule and the chain rule. The quotient rule states:
[tex]\[
\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\][/tex]
where [tex]\(u = (1 + 3x)^7\)[/tex] and [tex]\(v = x^9\)[/tex].
2. Find [tex]\(\frac{du}{dx}\)[/tex] and [tex]\(\frac{dv}{dx}\)[/tex]:
[tex]\[
u = (1 + 3x)^7 \quad \Rightarrow \quad \frac{du}{dx} = 7(1 + 3x)^6 \cdot 3 = 21(1 + 3x)^6
\][/tex]
[tex]\[
v = x^9 \quad \Rightarrow \quad \frac{dv}{dx} = 9x^8
\][/tex]
3. Apply the quotient rule:
[tex]\[
\frac{dy}{dx} = \frac{x^9 \cdot 21(1+3x)^6 - (1+3x)^7 \cdot 9x^8}{(x^9)^2}
\][/tex]
Simplify the numerator:
[tex]\[
= \frac{21x^9(1+3x)^6 - 9x^8(1+3x)^7}{x^{18}}
\][/tex]
Factor out [tex]\((1+3x)^6 x^8\)[/tex] from the numerator:
[tex]\[
= \frac{3x^8 (1+3x)^6 (7x - 3(1+3x))}{x^{18}}
= \frac{3(1+3x)^6(7x - 3 - 9x)}{x^{10}}
= \frac{3(1+3x)^6(-2x - 3)}{x^{10}}
= -\frac{6x + 9}{x^{10}}
\][/tex]
4. Evaluate [tex]\(\frac{dy}{dx}\)[/tex] at the point [tex]\(x = 1\)[/tex]:
[tex]\[
\frac{dy}{dx} = \left( -\frac{6(1) + 9}{(1)^{10}} \right) = -15
\][/tex]
5. Verify the derived value:
However, there was a misstep in the differentiation process. Using a more precise computation, the correct derivative at [tex]\(x = 1\)[/tex] is instead:
[tex]\[
\left. \frac{dy}{dx} \right|_{x=1} = -61440
\][/tex]
Thus, [tex]\(\frac{dy}{dx}\)[/tex] at the point [tex]\((1, 1)\)[/tex] for the given function is [tex]\(-61440\)[/tex].
Since none of the provided multiple-choice answers match [tex]\(-61440\)[/tex], it appears there may be a mistake in the options or the correct differentiation process yielding an unexpected result.
Given the numerical value derived from the solution steps, your answer would be:
[tex]\[
\boxed{-61440}
\][/tex]
However, please note that [tex]\(-1\)[/tex], [tex]\(11\)[/tex], [tex]\(5\)[/tex], and [tex]\(-\frac{15}{4}\)[/tex] do not represent the actual result derived.
