Second Order Derivatives (AQA A-Level Mathematics): Revision Notes

If $\ f(x)$ is a function, its first derivative $\ f'(x)$ represents the rate of change of $\ f(x)$ with respect to $\ x .$ The second derivative $\ f''(x)$ is the derivative of $\ f'(x) \ ,$ representing the rate of change of the first derivative.

• Mathematically: $f''(x) = \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d^2y}{dx^2}$ Here, $\ f''(x) \ or \ \frac{d^2y}{dx^2}$ denotes the second derivative of $\ y = f(x)$ with respect to $\ x \ .$

1. Find the First Derivative: Start by calculating the first derivative $\ f'(x)$ of the function $\ f(x) \ .$
2. Differentiate Again: Take the derivative of $\ f'(x)$ to obtain the second derivative $\ f''(x)$.

Example 1: Find the second derivative of $\ f(x) = x^3 - 3x^2 + 2x$.

• Step 1: Find the first derivative: $f'(x) = \frac{d}{dx}(x^3 - 3x^2 + 2x) = 3x^2 - 6x + 2$
• Step 2: Find the second derivative: $f''(x) = \frac{d}{dx}(3x^2 - 6x + 2) = 6x - 6$
• Interpretation:
• The second derivative $\ f''(x) = 6x - 6$ tells us about the concavity of the function $\ f(x)$. The sign of $\ f''(x)$ will determine where the function is concave up or concave down.
• $\ f''(x) > 0$ when $\ x > 1$, indicating the function is concave up for $\ x > 1 .$
• $\ f''(x) < 0$ when $\ x < 1$, indicating the function is concave down for $\ x < 1$.
• There is an inflexion point at $\ x = 1$, where the concavity changes.

Example 2: Find the inflexion point(s) of $\ f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 .$

• Step 1: Find the first derivative: $f'(x) = 4x^3 - 12x^2 + 12x - 4$
• Step 2: Find the second derivative: $f''(x) = 12x^2 - 24x + 12$
• Step 3: Set $\ f''(x) = 0$ to find potential inflexion points: $12x^2 - 24x + 12 = 0$ $x^2 - 2x + 1 = 0$ $(x - 1)^2 = 0$ $x = 1$
• Step 4: Check the sign of $\ f''(x) \ around \ x = 1$:
• The second derivative $\ f''(x) = 12(x - 1)^2$ is non-negative for all$\ x$, meaning it is $0$ at $\ x = 1$ and positive elsewhere. There is no change in concavity at $\ x = 1 \ , so \ x = 1$ is not an inflexion point.
• Conclusion:
• Since $\ f''(x)$does not change sign at $\ x = 1 \ ,$ $f(x)$ does not have an inflexion point.

• Second order derivatives provide valuable information about the curvature of a function, indicating where the function is concave up or concave down.
• Inflection points occur where the second derivative changes sign, marking a change in the concavity of the function.
• The second derivative is widely used in optimisation, physics, and economics to analyse and predict the behaviour of functions and systems.