Section 5

Higher-order derivatives

Given \(y = f(x)\), which is continuous and differentiable on its domain
\(\color{red}{{f}'(x)}\) is

the first-order derivative    
the slope of the function \(f(x)\)
the marginal function      

\(\color{red}{{f}''(x) = \frac{d}{dx}{f}'(x)}\) is

the second-order derivative 
the slope of the first derivative
  the slope of the marginal function

\(\color{red}{{f}'''(x)} = \frac{d}{dx}{f}''(x)\) is the third-order derivative.
And so on.

Second-order derivatives are important to test the concavity of the functions, to find whether an extreme point is a max, min or neither, to see the point where the concavity of the function changes, etc.