Calculating Higher-Order Derivatives

Differentiate the derivative

• Nothing stops you from differentiating $f'$ again. The result is the second derivative 二阶导数 $f''$.
• Keep going and you get higher-order derivatives 高阶导数: $f'''$, $f^{(4)}$, and so on.
• Each one measures the rate of change of the one before it.
• The second derivative is the star: it describes how the slope itself is changing.

Just repeat the process

• Find $f'$ using any rules you need, then differentiate that to get $f''$.
• $f(x)=x^4 \Rightarrow f'(x)=4x^3 \Rightarrow f''(x)=12x^2 \Rightarrow f'''(x)=24x$.
• Each step is an ordinary derivative — no new technique, just applied again.
• Simplify $f'$ before differentiating again; it keeps the algebra clean.

A cubic with its first and second derivatives on the same x-axis

Notation for the higher orders

• Prime notation: $f''(x)$, $f'''(x)$, then $f^{(4)}(x)$ (numbers past three primes).
• Leibniz notation: $\dfrac{d^2y}{dx^2}$ for the second derivative, $\dfrac{d^3y}{dx^3}$ for the third.
• Read $\dfrac{d^2y}{dx^2}$ as "d-squared-y d-x-squared" — it means "differentiate $y$ twice."
• The two notations mean exactly the same thing.

What the second derivative means

• $f'$ is the rate of change of $f$; $f''$ is the rate of change of $f'$.
• If $f$ is position, then $f'$ is velocity and $f''$ is acceleration — how fast the velocity changes.
• The sign of $f''$ tells you about concavity (curving up or down), coming up in Unit 5.
• So higher-order derivatives aren't busywork — $f''$ carries real, physical meaning.