Final answer:
To find y'', use implicit differentiation. Differentiate both sides of the equation with respect to x. Then, differentiate the resulting equation again with respect to x. Solve the resulting equation for y'' to find the second derivative.
Step-by-step explanation:
To find y'', you can use implicit differentiation. Start by differentiating both sides of the equation with respect to x.
For 5x² - y² = 6, the derivative of 5x² with respect to x is 10x, and the derivative of y² with respect to x is 2y * y', where y' represents dy/dx.
Therefore, the equation becomes 10x - 2y * y' = 0. To find y'', differentiate this equation again with respect to x. The derivative of 10x with respect to x is 10, and the derivative of -2y * y' with respect to x is -2y' * y' - 2y * y''.
So, the final equation is 10 - 2y' * y' - 2y * y'' = 0. Solve this equation for y'' to find the second derivative.