Higher-Order Derivatives

The first derivative, $f'(x)$, describes the instantaneous rate of change, or the slope, of a function $f(x)$ at any given point $x$. This slope indicates whether the function is increasing or decreasing. But what if there's a need to understand how the slope itself is changing? For this purpose, higher-order derivatives are used.

The Second Derivative: Measuring Curvature

Just as we can differentiate a function $f(x)$ to get its derivative $f'(x)$, we can differentiate the derivative $f'(x)$ again. The result is called the second derivative of $f(x)$.

Common notations for the second derivative include:

• $f''(x)$ (read as "f double prime of x")
• $\frac{d^2y}{dx^2}$ (read as "d squared y by dx squared")
• $y''$

The second derivative measures the rate at which the first derivative (the slope) changes. Think about driving a car:

• Your position corresponds to $f(x)$.
• Your velocity (speed and direction) corresponds to the first derivative, $f'(x)$.
• Your acceleration (how quickly your velocity is changing) corresponds to the second derivative, $f''(x)$.

In the context of a function's graph, the second derivative tells us about its concavity.

• Concave Up: If $f''(x) > 0$ on an interval, the slope $f'(x)$ is increasing. The graph bends upwards, like a cup holding water.
• Concave Down: If $f''(x) < 0$ on an interval, the slope $f'(x)$ is decreasing. The graph bends downwards, like an overturned cup or a dome.
• Inflection Point: A point where the concavity changes (from up to down, or down to up) is called an inflection point. At such a point, we often find $f''(x) = 0$, although $f''(x) = 0$ doesn't guarantee an inflection point.

Consider the function $f(x) = x^3$. Its first derivative is $f'(x) = 3x^2$. Its second derivative is $f''(x) = 6x$.

• For $x > 0$, $f''(x) > 0$, so the graph of $f(x)=x^3$ is concave up.
• For $x < 0$, $f''(x) < 0$, so the graph of $f(x)=x^3$ is concave down.
• At $x = 0$, $f''(x) = 0$. This point $(0,0)$ is an inflection point where the concavity changes.

The function $f(x)=x^3$ changes from concave down ($f''(x)<0$) to concave up ($f''(x)>0$) at $x=0$. Notice how the slope $f'(x)$ decreases until $x=0$ and then increases.

Derivatives After the Second

We can continue this process. Differentiating the second derivative $f''(x)$ gives the third derivative, denoted $f'''(x)$ or $\frac{d^3y}{dx^3}$. Differentiating again gives the fourth derivative, $f^{(4)}(x)$ or $\frac{d^4y}{dx^4}$, and so on. The derivative of $f(x)$ taken $n$ times is the nth derivative, denoted $f^{(n)}(x)$ or $\frac{d^ny}{dx^n}$.

Let's find the first few derivatives of a polynomial: $f(x) = 2x^4 - 5x^3 + x^2 - 7x + 3$

1. First Derivative (Slope): $f'(x) = \frac{d}{dx}(2x^4 - 5x^3 + x^2 - 7x + 3)$ $f'(x) = 8x^3 - 15x^2 + 2x - 7$

2. Second Derivative (Concavity): $f''(x) = \frac{d}{dx}(8x^3 - 15x^2 + 2x - 7)$ $f''(x) = 24x^2 - 30x + 2$

3. Third Derivative (Rate of change of concavity): $f'''(x) = \frac{d}{dx}(24x^2 - 30x + 2)$ $f'''(x) = 48x - 30$

4. Fourth Derivative: $f^{(4)}(x) = \frac{d}{dx}(48x - 30)$ $f^{(4)}(x) = 48$

5. Fifth Derivative (and higher): $f^{(5)}(x) = \frac{d}{dx}(48)$ $f^{(5)}(x) = 0$ All subsequent derivatives will also be zero.

Why Higher-Order Derivatives Matter

While the first derivative tells us the direction of change (is the function increasing or decreasing?), the second derivative provides significant information about the shape of the function's graph. This is particularly useful in optimization.

In the next section, we'll see how combining the first derivative (to find potential flat spots, where $f'(x)=0$) with the second derivative (to check the curvature at those spots) allows us to reliably identify local minima and maxima. This is known as the Second Derivative Test.

• If $f'(c) = 0$ and $f''(c) > 0$, the function is concave up at $x=c$, indicating a local minimum.
• If $f'(c) = 0$ and $f''(c) < 0$, the function is concave down at $x=c$, indicating a local maximum.

Understanding curvature through the second derivative is fundamental for analyzing optimization problems in machine learning. While we primarily use the first derivative (gradient) in algorithms like gradient descent, the concept of curvature (captured by the second derivative and its multivariable counterpart, the Hessian matrix) helps explain the behavior of these algorithms and motivates more advanced optimization techniques.