Final answer:
To find y' and y'' by implicit differentiation, differentiate both sides of the equation with respect to x. The derivative of x² is 2x, and the derivative of 5y² is 10yy'. So, y' = x / (5y) and y'' = (5y - x * y') / (5y)².
Step-by-step explanation:
To find y' and y'' by implicit differentiation, we will differentiate both sides of the equation with respect to x. Let's start with the equation x² - 5y² = 5. The derivative of x² is 2x, and the derivative of 5y² is 10yy'. The derivative of 5 is 0. So, the equation becomes 2x - 10yy' = 0.
To find y', we solve the equation for y': y' = 2x / (10y). Simplifying further, we get y' = x / (5y).
To find y'', we need to differentiate y' with respect to x. Using the quotient rule, we get y'' = (5y - x * y') / (5y)².