Chapter 3.1, Problem 38E

To determine

The correct value of $f\text{'}\left(-1\right)$ from (A) $-7$ (B) $7$ (C) $-3$ (D) $3$ (E) does not exist.

Answer to Problem 38E

The correct answer is (C)

Explanation of Solution

Given information:

The function: $f\left(x\right)=4-3x$

Formula Used:

Linear equation form:

  $y=mx+c$

Here $m$ represent the slop and $c$ represent the y-intersect.

Power rule of derivative.

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

Where $x$ is the variables and $n$ is constant term.

Calculation:

The given function is $f\left(x\right)=4-3x$ .

Now, find the derivative

Differentiate each side with respect to $x$ , use the power rule of derivative.

  $\begin{array}{c}\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(-3x+4\right)\\ =\frac{d}{dx}\left(-3x\right)+\frac{d}{dx}4\\ =-3\end{array}$

Given that the function is $f\left(x\right)=4-3x$ .

Rewrite the equation.

  $f\left(x\right)=-3x+4$

Compare this equation with the linear equation form $y=mx+c$ .

Thus, the function $f\left(x\right)$ represent a linear function.

Therefore, Slop $=-3$ , and the y-intersect $=4$ .

Thus, $f\text{'}\left(x\right)=-3$ for all $x$ .

Hence, the correct answer is (C)