Question
Watch Video

If $-x y+1+x=0$ then find $\frac{d^{2} y}{d x^{2}}$ at the point $(1,2)$ in simplest form.

Step 1 ：Given the equation \(-x y+1+x=0\), we are asked to find the second derivative of y with respect to x at the point (1,2).

Step 2 ：First, we rearrange the equation to express y in terms of x, giving us \(y = \frac{x + 1}{x}\).

Step 3 ：Next, we find the first derivative of y with respect to x, which is \(\frac{dy}{dx} = \frac{1}{x} - \frac{x + 1}{x^{2}}\).

Step 4 ：Then, we find the second derivative of y with respect to x, which is \(\frac{d^{2}y}{dx^{2}} = -\frac{2}{x^{2}} + \frac{2(x + 1)}{x^{3}}\).

Step 5 ：Finally, we evaluate the second derivative at the point (1,2), which gives us a value of 2.

Step 6 ：So, the second derivative of y with respect to x at the point (1,2) for the given equation is \(\boxed{2}\).