The derivative of a function is another function (Constant, Power and Sum Rules makes this concrete), so we can differentiate it again. The second derivative measures how the slope itself changes.
Key idea
$f''(x)$ is the derivative of $f'(x)$; $f'''(x)$ is the derivative of $f''$; and $f^{(n)}$ is the $n$th derivative. Leibniz notation: $\dfrac{d^2y}{dx^2}$, $\dfrac{d^3y}{dx^3}$. If $s(t)$ is position, $s'$ is velocity and $s''$ is acceleration.
Why it works
The first derivative is a rate of change. Its rate of change says whether that rate is speeding up or slowing down. For motion this is acceleration. For graphs it describes bending: when $f'' > 0$ the slopes are increasing, so the graph bends upward like a cup. The units multiply out the same way: if $s$ is in meters and $t$ in seconds, $s''$ is in meters per second per second.
Worked example: Repeat the power rule
Find $f''$ and $f'''$ for $f(x) = x^4 - 3x^2 + 5$.
- $f'(x) = 4x^3 - 6x$.
- $f''(x) = 12x^2 - 6$.
- $f'''(x) = 24x$.
Answer: $f''(x) = 12x^2 - 6$, $f'''(x) = 24x$
Worked example: A pattern
Find a formula for the $n$th derivative of $f(x) = \frac{1}{x}$.
- $f = x^{-1}$, $f' = -x^{-2}$, $f'' = 2x^{-3}$, $f''' = -6x^{-4}$, $f^{(4)} = 24x^{-5}$.
- Each step multiplies by the current exponent ($-1, -2, -3, \ldots$), so the signs alternate and the coefficients are $1, 2, 6, 24 = n!$.
Answer: $f^{(n)}(x) = \frac{(-1)^n\,n!}{x^{n+1}}$
Worked example: Acceleration
A car's position is $s(t) = 2t^3 - 9t^2 + 12t$ meters. Find its acceleration at $t = 2$ seconds.
- $v(t) = s'(t) = 6t^2 - 18t + 12$.
- $a(t) = v'(t) = 12t - 18$.
- $a(2) = 24 - 18 = 6$.
Answer: $6$ m/s²
In the next unit, the sign of $f''$ will tell us which way a graph bends (concavity) and will classify peaks and valleys. A positive second derivative means the slopes are increasing: the graph curves upward like a cup. A negative one means it curves downward like a cap.
For a polynomial of degree $n$, the $n$th derivative is a constant and every derivative after that is 0. For sine and cosine the derivatives repeat in a cycle of four: $\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x$.
Common mistake
Squaring the first derivative instead of differentiating twice: $f''(x)$ is not $\big(f'(x)\big)^2$. For $f(x) = x^3$: $f''(x) = 6x$, but $(f'(x))^2 = 9x^4$. Check the notation: the primes count derivatives, not powers.