Q. 15 - Calculus

Question

Express the sentence “the derivative of $f\left(x\right)=3x^2-1$ at $x=-4$ is $-24$” in each of the following notations:

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

Step-by-Step Solution

Answer

The derivative of $f\left(x\right)=3x^2-1$ at $x=-4$ $-24$ in following notations is:

(a) $$\begin{gathered} \underset{x\to -4}{\lim }\frac{f\left(x\right)-f\left(-4\right)}{x-\left(-4\right)}=\underset{∆x\to 0}{\lim }\frac{∆y}{∆x} \\ =-24 \end{gathered}$$

(b) ${\overline{)\frac{dy}{dx}}}_{x=-4}=-24$

(c) ${\overline{)\frac{d}{dx}\left[f\left(x\right)\right]}}_{x=-4}=-24$

Step 1. Given information

We have been given the derivative of $f\left(x\right)=3x^2-1$ at $x=-4$ is $-24$.

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

$$\begin{gathered} \underset{x\to -4}{\lim }\frac{f\left(x\right)-f\left(-4\right)}{x-\left(-4\right)}=\underset{∆x\to 0}{\lim }\frac{∆y}{∆x} \\ =-24 \end{gathered}$$

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

${\overline{)\frac{dy}{dx}}}_{x=-4}=-24$

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

${\overline{)\frac{d}{dx}\left[f\left(x\right)\right]}}_{x=-4}=-24$