Notation of derivatives
The derivative of a function \(f\) is a function itself. The term ‘taking the derivative’ of ‘differentiating’ means to find the expression representing the derivative, which is then denoted by one of two commonly used ways: Lagrange notation (also called ‘prime’ notation) and Leibniz notation.
For the purpose of this summary, we will consider the function \[y=f(x)=x^2\]
| Lagrange notation | Prime notation | |
|---|---|---|
| First derivative | \(y\color{red}{'}=f\color{red}{'}(x)=(x^2)\color{red}{'}\) The \(\color{red}{'}\) is written on the name of the function (\(y\) or \(f\)) or on the whole parenthesized expression. |
\(\dfrac{\color{red}{d}y}{\color{red}{dx}}=\color{red}{\dfrac{d}{dx}}f(x)=\color{red}{\dfrac{d}{dx}}x^2\) The name of the function is written in the ‘numerator’ only if it is a single letter (like \(y\)). The notation does not represent a quotient, but the limit of a quotient. |
| Differentiation operator | The ‘prime‘ symbol \(\color{red}{'}\) means we are taking the derivative | The symbol \(\color{red}{\dfrac{d}{dx}}\) means we are taking the derivative with respect to \(x\) |
| Pros | Simple and concise | Explicitly states the variable of differentiation (\(x\) in this case) |
| Cons | Does not explicitly state the variable of differentiation. Get quickly confusing in more complex scenarios. |
More complicated to use in trivial cases (like functions of single variables) |
| Second derivative | \(\begin{aligned} \left(y'\right)'&=y'' \\ \left[f'(x)\right]'&=f''(x) \\ \left((x^2)'\right)'&=(x^2)'' \end{aligned}\) | \(\begin{aligned} \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)&=\dfrac{d^2y}{dx^2} \\ \dfrac{d}{dx}\left(\dfrac{d}{dx}f(x)\right)&=\dfrac{d^2}{dx^2}f(x) \\ \dfrac{d}{dx}\left(\dfrac{d}{dx}x^2\right)&=\dfrac{d^2}{dx^2}x^2 \end{aligned}\) |
| Third derivative | \(\begin{aligned} \left(y''\right)'&=y''' \\ \left[f''(x)\right]'&=f'''(x) \\ \left((x^2)''\right)'&=(x^2)''' \end{aligned}\) | \(\begin{aligned} \dfrac{d}{dx}\left[\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)\right]&=\dfrac{d^3y}{dx^3} \\ \dfrac{d}{dx}\left[\dfrac{d}{dx}\left(\dfrac{d}{dx}f(x)\right)\right]&=\dfrac{d^3}{dx^3}f(x) \\ \dfrac{d}{dx}\left[\dfrac{d}{dx}\left(\dfrac{d}{dx}x^2\right)\right]&=\dfrac{d^3}{dx^3}x^2 \end{aligned}\) |
| Higher derivatives | \(y^{(n)}=f^{(n)}(x)=(x^2)^{(n)}\) Parentheses around the \(n\) are not optional; otherwise the notation is confused with exponentiation. |
\(\dfrac{d^ny}{dx^n}=\dfrac{d^n}{dx^n}f(x)=\dfrac{d^n}{dx^n}x^2\) |
| The number \(n\in\mathbb{N}\) is called the order of the derivative. | ||