Notes on Second Derivatives and Concavity

Higher-Order Derivatives

When we differentiate a function, $f(x)$, we obtain its first derivative, denoted as $f'(x)$. It is possible to differentiate the derivative itself. When we differentiate the first derivative, $f'(x)$, we obtain the second derivative of $f(x)$. This is denoted as $f''(x)$, often read as "f double prime."

Notation

• Prime Notation:

  • First derivative: $f'(x)$
  • Second derivative: $f''(x)$
  • Third derivative: While one might say $f'''(x)$, it's generally preferred to use numerical superscripts for higher derivatives, such as $f^{(3)}(x)$.

• Leibniz Notation:

  • First derivative: $\frac{dy}{dx}$
  • Second derivative: $\frac{d^2y}{dx^2}$ (Note: This notation does not imply squaring anything; it is simply the established notation for the second derivative).

It is possible to continue this process to find higher-order derivatives (third, fourth, and so on), as long as the function remains differentiable. For higher derivatives, the prime notation can become unwieldy, so we often use a numerical superscript in parentheses, e.g., $f^{(n)}(x)$ for the $n$-th derivative.

Relationship Between Second Derivative and First Derivative

The second derivative tells us about the behavior of the first derivative:

• If the second derivative is positive (f''(x) > 0), then the first derivative ($f'(x)$) is increasing. This means the slope of the original function is getting larger (moving from more negative to less negative, or from less positive to more positive).
• If the second derivative is negative (f''(x) < 0), then the first derivative ($f'(x)$) is decreasing. This means the slope of the original function is getting smaller (moving from more positive to less positive, or from less negative to more negative).

Concavity

The concept of an increasing or decreasing first derivative leads us directly to concavity, which describes the