Higher-Order Derivatives

When you take the derivative of a function $f(x)$, the result is a new function, $f'(x)$. Because $f'(x)$ is also a function, you can take its derivative as well. This process of differentiating a derivative yields what are known as higher-order derivatives.

Notation

The derivative of the first derivative is called the second derivative. If you differentiate again, you get the third derivative, and so on.

Here is the common notation used for higher-order derivatives:

• First derivative: $f'(x)$ or $\frac{dy}{dx}$
• Second derivative: $f''(x)$ or $\frac{d^2y}{dx^2}$
• Third derivative: $f'''(x)$ or $\frac{d^3y}{dx^3}$
• $n$-th derivative: $f^{(n)}(x)$ or $\frac{d^ny}{dx^n}$

(Note: After the third derivative, we typically use numbers in parentheses to avoid writing too many prime marks.)

Physical Interpretation (Kinematics)

Higher-order derivatives are incredibly useful in physics, specifically for describing motion. If a function $s(t)$ represents the position of an object over time $t$:

• Velocity $v(t)$ is the rate of change of position (the first derivative): $v(t) = s'(t)$
• Acceleration $a(t)$ is the rate of change of velocity (the second derivative of position): $a(t) = v'(t) = s''(t)$
• Jerk $j(t)$ is the rate of change of acceleration (the third derivative of position): $j(t) = a'(t) = s'''(t)$

Geometric Interpretation (Concavity)

Graphically, higher-order derivatives tell us about the shape of the original function's curve.

• First Derivative ($f'$): Tells us the slope of the tangent line. It shows where the function is increasing ($f' > 0$) or decreasing ($f' < 0$).
• Second Derivative ($f''$): Tells us the concavity of the function, which is how the curve bends.
  • If $f''(x) > 0$, the graph is concave up (shaped like a cup, $\cup$).
  • If $f''(x) < 0$, the graph is concave down (shaped like a frown, $\cap$).

Example Problems

Example 1: Finding the second derivative

Problem: Find $f''(x)$ for $f(x) = x^4 - 3x^2 + 2x$.

Solution: Step 1: Find the first derivative using the power rule. $f'(x) = 4x^3 - 6x + 2$

Step 2: Differentiate $f'(x)$ to find the second derivative. $f''(x) = 12x^2 - 6$

Example 2: Velocity, Acceleration, and Jerk

Problem: The position of a particle is given by $s(t) = t^3 - 6t$. Find the functions for velocity, acceleration, and jerk.

Solution:

• Velocity (first derivative): $v(t) = s'(t) = 3t^2 - 6$
• Acceleration (second derivative): $a(t) = s''(t) = 6t$
• Jerk (third derivative): $j(t) = s'''(t) = 6$