5. find each derivative.\na. $f(x)=x^{3}-7x + 5$\nb. $f(x)=x^{3}-7x - 20$\nc. $f(x)=x^{3}-7x + 11$

Explanation:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$, and the derivative of a constant $C$ is $0$, i.e., $(C)^\prime = 0$, and $(ax)^\prime=a$ for a constant $a$. Also, if $y = u\pm v\pm w$, then $y^\prime=u^\prime\pm v^\prime\pm w^\prime$.

Step2: Find derivative of $f(x)=x^{3}-7x + 5$

For $y = x^{3}-7x + 5$, let $u=x^{3}$, $v = - 7x$, $w = 5$. Then $u^\prime=3x^{2}$ (by power - rule with $n = 3$), $v^\prime=-7$ (since $(ax)^\prime=a$ with $a=-7$), and $w^\prime = 0$ (since derivative of a constant is $0$). So $f^\prime(x)=3x^{2}-7$.

Step3: Find derivative of $f(x)=x^{3}-7x - 20$

Let $u=x^{3}$, $v=-7x$, $w = - 20$. Then $u^\prime=3x^{2}$, $v^\prime=-7$, $w^\prime = 0$. So $f^\prime(x)=3x^{2}-7$.

Step4: Find derivative of $f(x)=x^{3}-7x + 11$

Let $u=x^{3}$, $v=-7x$, $w = 11$. Then $u^\prime=3x^{2}$, $v^\prime=-7$, $w^\prime = 0$. So $f^\prime(x)=3x^{2}-7$.

Answer:

a. $f^\prime(x)=3x^{2}-7$ b. $f^\prime(x)=3x^{2}-7$ c. $f^\prime(x)=3x^{2}-7$