Why is the second derivative written as d² $f/dx$ ² instead of a longer expression like d(df/dx)/dx?

Q: Why is the second derivative written as d²f/dx² instead of a longer expression like d(df/dx)/dx?

The notation $\frac{d^2 f}{dx^2}$ is a standard shorthand for the expanded differential form $\frac{d(\frac{df}{dx})}{dx}$. The expanded form literally means taking the differential of the first derivative and dividing by the differential of $x$. The compact notation avoids clutter while preserving the meaning of applying the differentiation operator twice.

The notation

\frac{d^2 f}{dx^2}

is a standard shorthand for the expanded differential form

\frac{d(\frac{df}{dx})}{dx}

. The expanded form literally means taking the differential of the first derivative and dividing by the differential of

x

. The compact notation avoids clutter while preserving the meaning of applying the differentiation operator twice.

Conditions

The function $f$ is twice differentiable.
Leibniz notation conventions are being used.

Reasoning, step by step

Start with the definition of the second derivative as the derivative of the first derivative.
Write the first derivative as $\frac{df}{dx}$ .
Apply the differential operator $d$ to $\frac{df}{dx}$ and divide by $dx$ , yielding $\frac{d(\frac{df}{dx})}{dx}$ .
Recognize that this expanded form is cumbersome.
Adopt the conventional abbreviation $\frac{d^2 f}{dx^2}$ for simplicity.

Example

The video shows the transition from $\frac{d(\frac{df}{dx})}{dx}$ to $\frac{d^2 f}{dx^2}$ , explaining that the latter is the standard way to write the former.

Common misconceptions

Thinking that $d^2$ means squaring the variable $d$ algebraically.
Believing the notation implies multiplying fractions rather than iterating operators.

Watch the explanation

BilibiliHigher order derivatives | 3Blue1Brown

1:56 – 2:23Watch this moment ↗

Connected concepts

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Why is the second derivative written as d² $f/dx$ ² instead of a longer expression like d(df/dx)/dx?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

Why is the second derivative written as d²f/dxf/dxf/dx² instead of a longer expression like d(df/dx)/dx?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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Why is the second derivative written as d² $f/dx$ ² instead of a longer expression like d(df/dx)/dx?