Q. 16 - Calculus

Question

Express the sentence “the fourth derivative of $f\left(x\right)=3x^2-1$ is equal to $0$” in each of the following notations:

(a) prime (b) Leibniz (c) operator

Step-by-Step Solution

Answer

The fourth derivative of $f\left(x\right)=3x^2-1$ is equal to $0$ in following notations is:

(a) $\underset{∆x\to 0}{\lim }\left(\underset{∆x\to 0}{\lim }\left(\underset{∆x\to 0}{\lim }\left(\underset{∆x\to 0}{\lim }\frac{∆y}{∆x}\right)\right)\right)=0$

(b) $\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right)\right)=0$

(c)  $\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}\left(3x^2-1\right)\right)\right)\right)=0$

Step 1. Given information

We have been given the fourth derivative of $f\left(x\right)=3x^2-1$ is equal to $0$.

Step 2. Part (a) Write the fourth derivative in a prime notation.

$\underset{∆x\to 0}{\lim }\left(\underset{∆x\to 0}{\lim }\left(\underset{∆x\to 0}{\lim }\left(\underset{∆x\to 0}{\lim }\frac{∆y}{∆x}\right)\right)\right)=0$

Step 3. Part (b) Write the fourth derivative in a Leibniz notation.

$\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right)\right)=0$

Step 4. Part (c) Write the fourth derivative in a operator notation.

$\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{d}{dx}\left(3x^2-1\right)\right)\right)\right)=0$