How do you calculate the second order derivative d²y/dx² of a function y = f(x)?

Calculating the second order derivative of a function involves a few clear steps. Here’s how to do it:

1. Find the First Derivative (dy/dx):

   The first step in finding the second order derivative is to determine the first derivative of the function. For a function y = f(x), you can use standard differentiation rules such as the power rule, product rule, quotient rule, or chain rule.

2. Differentiate Again:

   Once you have the first derivative dy/dx, the next step is to differentiate this result to find the second derivative. This means you will apply the differentiation rules again to dy/dx.

3. Express the Second Derivative:

   The result of this second differentiation gives you d²y/dx², which is the second order derivative of the original function y = f(x).

4. Example:

   Let’s consider the function y = x² + 3x + 2. First, we find the first derivative:

   dy/dx = 2x + 3

   Now, we differentiate again to find the second derivative:

   d²y/dx² = d/dx(2x + 3) = 2

   So, the second derivative of the function y = x² + 3x + 2 is

   d²y/dx² = 2.

In summary, to find the second order derivative d²y/dx² of any function y = f(x), simply differentiate the function twice in succession. This process lets you analyze the curvature and acceleration characteristics of the original function’s graph.