Second Order Derivative

5.7 Second Order Derivative

From a Rate to a Rate of Change

For $y = f(x)$, the first derivative $\dfrac{dy}{dx} = f'(x)$ tells us how fast $y$ changes as $x$ changes. But $f'(x)$ is itself just another function of $x$ — so we can ask the same question about it: how fast is $f'(x)$ changing? Differentiating $f'(x)$ once more with respect to $x$ answers that. On the left-hand side this is written as

and the result is called the second order derivative of $y$ with respect to $x$.

The Four Notations

You will see the second derivative written in several equivalent ways across textbooks and exam papers — they all mean the same thing:

• $\dfrac{d^2y}{dx^2}$ — the most common form, built by applying $\dfrac{d}{dx}$ twice
• $f''(x)$ — when $y = f(x)$, read as "f double prime of x"
• $D^2y$ — using the operator notation $D = \dfrac{d}{dx}$
• $y''$ or $y_2$ — compact shorthand used when the variable is unambiguous

A Worked Micro-Example

To see the mechanics, take $y = x^3$.

1. First derivative: $\dfrac{dy}{dx} = 3x^2$.
2. Now differentiate that result again with respect to $x$: $\dfrac{d}{dx}(3x^2) = 6x$.
3. So $\dfrac{d^2y}{dx^2} = 6x$.

There is no new rule here — the second derivative just applies the ordinary rules of differentiation (power, product, chain) to the first derivative, as you would to any function of $x$. Simplify $\dfrac{dy}{dx}$ fully before differentiating it again; a messy first derivative makes the second pass far more error-prone.

What It Actually Tells You

If the first derivative measures a rate of change, the second derivative measures the rate of change of that rate — how quickly the rate itself is speeding up, slowing down, or reversing. The classic example is motion: if $s(t)$ is a particle's position at time $t$, $\dfrac{ds}{dt}$ is its velocity, and $\dfrac{d^2s}{dt^2}$ is its acceleration — the rate at which the velocity is changing. The same idea applies to any $y = f(x)$: the second derivative describes how the curve is bending, which is why it becomes the key tool for testing concavity and locating maxima/minima in the next chapter.

Higher Order Derivatives

Nothing special happens at the second derivative. If $\dfrac{d^2y}{dx^2}$ is itself differentiable, differentiating it again gives the third order derivative $\dfrac{d^3y}{dx^3}$ (or $y'''$), and repeating gives the fourth, fifth, and so on. In general, the $n$th order derivative comes from differentiating $y$ successively $n$ times, provided every intermediate derivative is itself differentiable.

Important Differential Relations Established …