Answer
$3(1+\frac{1}{x}) ^2 \cdot \frac{1)}{x^3}$.
Work Step by Step
To find the second derivative of the given function $y = (1 + \frac{1}{x})^3$, we can use the chain rule and the power rule for differentiation.
The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
The power rule states that the derivative of $x^n$ is $n \cdot x^{n-1}$.
Applying the chain rule, we differentiate the outer function $u^3$ with respect to $u$ to get $3u^2$.
Applying rule, we differentiate the inner function $1 + \frac{1}{x}$ with respect to $x$ to get $\frac{-1}{x^2}$.
Therefore, the first derivative of the given function is $3(1 + \frac{1}{x})^2 \cdot \frac{-1}{x^3}$.
