Second-Order Derivatives

Welcome back to KnowledgeKnot! In this article, we delve into the concept of second-order derivatives in calculus, their meaning, computation, and real-world applications. Let’s begin!

What is a Second-Order Derivative?

The second-order derivative of a function is the derivative of its first derivative. It measures the rate of change of the rate of change (i.e., how the slope of a curve changes). Denoted as:

f''(x) = \frac{d}{dx}\left[f'(x)\right] = \frac{d^2y}{dx^2}

Here, $f''(x)$ provides information about the concavity and acceleration of a function.

Physical Interpretation

In physics, the second derivative has a direct interpretation:

If $f(x)$ represents position, $f'(x)$ represents velocity, and $f''(x)$ represents acceleration.
The second derivative helps understand how quickly the rate of change (velocity) itself is changing.

How to Compute a Second-Order Derivative?

To find the second-order derivative, follow these steps:

Find the first derivative of the function, $f'(x)$ .
Differentiate $f'(x)$ to get $f''(x)$ .

Examples

Example 1: Basic Polynomial

Find the second derivative of $f(x) = 2x^3 - 5x^2 + 3x - 7$ .

Solution:

Step 1: Find the first derivative:

f'(x) = \frac{d}{dx} [2x^3 - 5x^2 + 3x - 7] = 6x^2 - 10x + 3

Step 2: Differentiate again to find the second derivative:

f''(x) = \frac{d}{dx} [6x^2 - 10x + 3] = 12x - 10

The second-order derivative is $f''(x) = 12x - 10$ .

Example 2: Trigonometric Function

Find the second derivative of $f(x) = \sin(x)$ .

Solution:

Step 1: Find the first derivative:

f'(x) = \cos(x)

Step 2: Differentiate again to find the second derivative:

f''(x) = -\sin(x)

The second-order derivative is $f''(x) = -\sin(x)$ .

Applications of Second-Order Derivatives

Concavity: Determines whether a curve is concave up or concave down:
- $f''(x) > 0$ : Concave up (smile shape).
- $f''(x) < 0$ : Concave down (frown shape).
Second Derivative Test: Used to determine local maxima or minima:
- If $f'(x) = 0$ and $f''(x) > 0$ , the function has a local minimum.
- If $f'(x) = 0$ and $f''(x) < 0$ , the function has a local maximum.
Physics: Used in motion analysis to find acceleration.

Practice Problem

Problem:

Find the second derivative of $f(x) = \ln(x)$ .

Solution:

First derivative:
$f'(x) = \frac{1}{x}$
Second derivative:
$f''(x) = \frac{d}{dx} \left[\frac{1}{x}\right] = -\frac{1}{x^2}$

The second-order derivative is $f''(x) = -\frac{1}{x^2}$ .