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.)

18.role="math" localid="1648544151216" ${\overline{)\frac{dy}{dx}}}_{x=2},{\overline{)\frac{df}{dx}}}_{x=2},{\overline{)\frac{d}{dx}\left(f\left(x\right)\right)}}_{x=2},\frac{dy}{dx}\left(f\left(2\right)\right)$

Short answer

The answers are as follow:

$$\begin{gathered} {\overline{)\frac{dy}{dx}}}_{x=2}=10 \\ {\overline{)\frac{df}{dx}}}_{x=2}=10 \\ {\overline{)\frac{d}{dx}\left(f\left(x\right)\right)}}_{x=2}=10 \\ \frac{dy}{dx}\left(f\left(2\right)\right)=10 \end{gathered}$$

Step-by-step solution

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 ${\overline{)\frac{dy}{dx}}}_{x=2},{\overline{)\frac{df}{dx}}}_{x=2},{\overline{)\frac{d}{dx}\left(f\left(x\right)\right)}}_{x=2},\frac{dy}{dx}\left(f\left(2\right)\right)$are equal to $f\text{'}\left(2\right)$.

$$\begin{gathered} f\text{'}\left(2\right)=3{\left(2\right)}^2-2 \\ =10 \end{gathered}$$

So,

$$\begin{gathered} {\overline{)\frac{dy}{dx}}}_{x=2}=10 \\ {\overline{)\frac{df}{dx}}}_{x=2}=10 \\ {\overline{)\frac{d}{dx}\left(f\left(x\right)\right)}}_{x=2}=10 \\ \frac{dy}{dx}\left(f\left(2\right)\right)=10 \end{gathered}$$