Elaborate the Steps to Find d²y/dx² (Second Order Derivative) of Any Function y = f(x)

To find the second order derivative of a function y = f(x), you need to follow these steps:

1. Find the First Derivative: Start by determining the first derivative of the function f(x) with respect to x. This is denoted as dy/dx or f'(x). You can use differentiation rules, such as the power rule, product rule, or quotient rule, depending on the form of your function.
2. Differentiate Again: Once you have the first derivative, take the derivative of that result to find the second derivative. This involves applying the same differentiation techniques used in the first step. The notation for the second derivative is d²y/dx² or f”(x).
3. Simplify: After calculating the second derivative, simplify the expression if possible. This helps in better understanding and further calculations.

Example: Let’s say our function is f(x) = x³ + 3x² + 5.

1. First derivative: f'(x) = 3x² + 6x.

2. Second derivative: f”(x) = 6x + 6.

So, the second derivative d²y/dx² = 6x + 6. This result gives you information about the curvature of the function and can be used to determine concavity and inflection points.