Question

determine each of the following, where (f(x)=2x^{3}+3x - 1).
a. (f(x)=)
b. (f(-2)=)

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $f(x)=2x^{3}+3x - 1$, the derivative of $2x^{3}$ is $2\times3x^{3 - 1}=6x^{2}$, the derivative of $3x$ is $3\times1x^{1 - 1}=3$, and the derivative of the constant $-1$ is $0$. So, $f^\prime(x)=6x^{2}+3$.

Step2: Evaluate $f^\prime(x)$ at $x=-2$

Substitute $x = - 2$ into $f^\prime(x)$. We have $f^\prime(-2)=6\times(-2)^{2}+3$. First, calculate $(-2)^{2}=4$. Then $6\times4 + 3=24 + 3=27$.

Answer:

a. $f^\prime(x)=6x^{2}+3$
b. $f^\prime(-2)=27$