a. find the derivative function f’ for the function f. b. find an equation of the line tangent to the graph…

Answer:

$4x + 3$

Explanation:

Step1: Recall power - rule

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For a function $y = ax^n+bx^m+\cdots$, the derivative $y^\prime=anx^{n - 1}+bmx^{m - 1}+\cdots$.

Step2: Differentiate each term

Given $f(x)=2x^{2}+3x - 2$. The derivative of $2x^{2}$ using the power - rule: If $y = 2x^{2}$, then $y^\prime=2\times2x^{2 - 1}=4x$. The derivative of $3x$: If $y = 3x$, then $y^\prime=3\times1x^{1 - 1}=3$. The derivative of the constant $-2$ is $0$ (since the derivative of a constant $C$ is $0$).

Step3: Combine the derivatives

$f^\prime(x)$ is the sum of the derivatives of each term. So $f^\prime(x)=4x + 3$.