Question

Leibniz notation: Describe the meanings of each of the mathematical expressions that follow. Translate expressions written in Leibniz notation to “prime” notation, and vice versa.

\(\frac{d}{dx}|_{-1}(x^{2})\)

Short answer

The notation are Lagrange’s notation \(\frac{d}{dx}|_{-1}(x^{2})=f'(x^{2})\) and Eular notation \(\frac{d}{dx}|_{-1}(x^{2})=D(x^{2})\) and \(D(x^{2})=f'(x^{2})=\frac{d}{dx}|_{-1}(x^{2})=-2\).

Step-by-step solution

Step 1. Given information

\(\frac{d}{dx}|_{-1}(x^{2})\) which is Leibniz’s notation

Concept used:the notation is different but the meaning is same out of these notation Leibniz’s notation is most commonly used in calculus.

Step 2. Calculation

Here, \(\frac{d}{dx}|_{-1}(x^{2})\) which is

In Lagrange's notation \(\frac{d}{dx}|_{-1}(x^{2})=f'(x^{2})\) at \(x=-1\)

In Euler notation \(\frac{d}{dx}|_{-1}(x^{2})=D(x^{2})\) at \(x=--1\)

Here \(x^{2}\) is function given here it changes with \(x\) putting \(x =-1\) after differentiation \(x^{2}\).

\(D(x^{2})=2x\)

\(\frac{d}{dx}|_{-1}(x^{2})=2x\)

\(f'(x^{2})=2x\)

So, by putting \(x=-1\) in the above function we get

\(D(x^{2})=f'(x^{2})=\frac{d}{dx}|_{-1}(x^{2})=-2\)

Hence,

The notation are Lagrange’s notation \(\frac{d}{dx}|_{-1}(x^{2})=f'(x^{2})\) and Eular notation \(\frac{d}{dx}|_{-1}(x^{2})=D(x^{2})\) and \(D(x^{2})=f'(x^{2})=\frac{d}{dx}|_{-1}(x^{2})=-2\).