Q. 17 - Calculus

Question

Suppose $f\left(x\right)=x^3-2x+1$, and let $y=f\left(x\right)$. It can be shown that $f\text{'}\left(x\right)=3x^2-2$. Use this information to determine the expressions in Exercises 17 and 18 (Note: No differentiation will be necessary, since the derivatives are given. These problems are just a test of your ability to interpret derivative notation.)

17. $\frac{dy}{dx}, \frac{df}{dx}, \frac{d}{dx}\left(f\left(x\right)\right), \frac{dy}{dx}\left(f\left(x\right)\right)$

Step-by-Step Solution

Answer

The answers are as follow:

$$\begin{gathered} \frac{dy}{dx}=3x^2-2 \\ \frac{df}{dx}=3x^2-2 \\ \frac{d}{dx}\left(f\left(x\right)\right)=3x^2-2 \\ \frac{dy}{dx}\left(f\left(x\right)\right)=3x^2-2 \end{gathered}$$

Step 1. Given information

We have been given

$$\begin{gathered} f\left(x\right)=x^3-2x+1 \\ y=f\left(x\right) \\ f\text{'}\left(x\right)=3x^2-2 \end{gathered}$$

Step 2. Write answers for different derivative notations.

All the notations $\frac{dy}{dx}, \frac{df}{dx}, \frac{d}{dx}\left(f\left(x\right)\right), \frac{dy}{dx}\left(f\left(x\right)\right)$ are equal to $f\text{'}\left(x\right)$ and $f\text{'}\left(x\right)=3x^2-2$.

So,

$$\begin{gathered} \frac{dy}{dx}=3x^2-2 \\ \frac{df}{dx}=3x^2-2 \\ \frac{d}{dx}\left(f\left(x\right)\right)=3x^2-2 \\ \frac{dy}{dx}\left(f\left(x\right)\right)=3x^2-2 \end{gathered}$$