use implicit differentiation to find the second derivative.

Question

x^2-y^2=1

Answer 1

First, let's find the first derivative using implicit differentiation. We have the equation:

x^2 - y^2 = 1

Differentiating both sides with respect to x, we get:

2x - 2y(dy/dx) = 0

Now, solve for dy/dx:

2y(dy/dx) = 2x

(dy/dx) = 2x/2y

(dy/dx) = x/y

Now, let's find the second derivative by differentiating dy/dx with respect to x:

d^2y/dx^2 = (d/dx)(x/y)

To differentiate the right side, we will use the quotient rule which states that:

(d/dx)(u/v) = (vu' - uv')/v^2

In this case, u = x and v = y. So we get:

u' = d/dx(x) = 1
v' = d/dx(y) = dy/dx (since we don't have an explicit expression for y, we will leave it as dy/dx)

Now, applying the quotient rule:

d^2y/dx^2 = (y(1) - x(dy/dx))/y^2

We already found (dy/dx) = x/y, so we can substitute it back in:

d^2y/dx^2 = (y - x(x/y))/y^2

Simplify:

d^2y/dx^2 = (y^2 - x^2)/(y^3)

So the second derivative is:

d^2y/dx^2 = (y^2 - x^2)/(y^3)