Second Order Derivatives

Definition: The second order derivative is obtained by differentiating the first derivative of a function $f(x)$ with respect to $x$. It represents how the slope of the original function is changing at any given point.

Notation: There are two primary ways to denote this operation. In functional notation, it is written as $f''(x)$, while in Leibniz notation, it is expressed as $\frac{d^2y}{dx^2}$.

Leibniz Syntax: In the notation $\frac{d^2y}{dx^2}$, the '2' indicates the order of the derivative. Note that the '2' is placed after the '$d$' in the numerator and after the '$x$' in the denominator.

Rate of Change of Gradient: Just as the first derivative $f'(x)$ gives the rate of change of $y$, the second derivative $f''(x)$ gives the rate of change of the gradient. If $f''(x)$ is positive, the gradient is increasing; if negative, the gradient is decreasing.

Concavity: The second derivative determines the 'hollow' of the curve. A positive second derivative indicates the curve is concave up (like a cup), while a negative value indicates the curve is concave down (like a cap).

Physical Interpretation: In kinematics, if $s$ is displacement and $t$ is time, the first derivative $\frac{ds}{dt}$ is velocity, and the second derivative $\frac{d^2s}{dt^2}$ is acceleration.

Step 1: First Differentiation: Begin by differentiating the original function $y = f(x)$ using standard rules (power rule, chain rule, etc.) to find $\frac{dy}{dx}$ or $f'(x)$.

Step 2: Second Differentiation: Differentiate the resulting expression from Step 1 again with respect to $x$. This yields $\frac{d^2y}{dx^2}$ or $f''(x)$.

Handling Complex Terms: When dealing with fractions or roots, always rewrite them as powers of $x$ (e.g., $\frac{1}{x^2}$ as $x^{-2}$) before differentiating to avoid algebraic errors.

Evaluation: To find the value of the second derivative at a specific point, substitute the $x$-coordinate of that point into the $f''(x)$ expression.

The Inconclusive Case: If you calculate $f''(x) = 0$ at a stationary point, do not assume it is a point of inflection. You must revert to checking the gradient ($f'(x)$) slightly to the left and right of the point to confirm its nature.

Notation Precision: Ensure you place the superscript '2' correctly in $\frac{d^2y}{dx^2}$. Examiners often look for this specific detail to verify mathematical literacy.

Verification: If a question asks you to 'show' that a point is a minimum, your final step must clearly state that $f''(x) > 0$ and therefore the point is a local minimum.

Chain Rule Awareness: When differentiating a second time, remember that the chain rule or product rule may apply again if the first derivative is a composite or product function.

Sign Confusion: A common mistake is thinking $f''(x) > 0$ means a maximum because 'positive' feels like 'top'. Remember: positive second derivative means the gradient is growing, which happens at the bottom of a 'cup' (minimum).

Algebraic Errors with Negatives: When differentiating terms with negative powers (like $x^{-1}$), the power becomes more negative ($-2$). Students often incorrectly 'increase' it to $0$.

Forgetting the Second Step: In multi-part problems, students sometimes find the first derivative and stop, forgetting that the 'nature' of a point requires the second derivative.