use implicit differentiation to find the 2nd derivative:
x^2-y^2=1
(pls do step by step, also when u solve for 1st deriv, you're to plug that answer for dy/dx when finding 2nd derivative)
To begin, we use implicit differentiation to find the first derivative:
Take the derivative of both sides of the equation with respect to x:
(d/dx)(x^2-y^2)=(d/dx)1
Using the power rule for derivatives:
2x - 2y(dy/dx) = 0
Rearranging for dy/dx (the first derivative):
dy/dx = 2x/2y = x/y
To find the second derivative, we differentiate the first derivative obtained above using implicit differentiation.
Take the derivative of both sides of the equation with respect to x:
(d/dx)(dy/dx) = (d/dx)(x/y)
Using the quotient rule for derivatives:
d^2y/dx^2 = (y(1)(2y) - x(dy/dx))/y^2
Substituting our expression for dy/dx from the first derivative:
d^2y/dx^2 = (y(1)(2y) - x(x/y))/y^2
Simplifying:
d^2y/dx^2 = (2y^2 - x^2)/y^3
Therefore, the second derivative of the function x^2 - y^2 = 1 with respect to x is (2y^2 - x^2)/y^3.