Evaluate the difference quotient for the function (f(x) = 3x^2 + 2x + 1).

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Step:1 ：Step 1: First, remember that the difference quotient is defined as [frac{f(x + h) - f(x)}{h}]Step:2 ：Step 2: Substitute (f(x + h)) and (f(x)) into the formula. [frac{(3(x + h)^2 + 2(x + h) + 1) - (3x^2 + 2x + 1)}{h}]Step:3 ：Step 3: Expand and simplify the numerator.[frac{(3x^2 + 6xh + 3h^2 + 2x + 2h + 1) - (3x^2 + 2x + 1)}{h}] [=frac{6xh + 3h^2 + 2h}{h}]Step:4 ：Step 4: Cancel out (h) from the numerator and denominator. [= 6x + 3h + 2]

Answer

Step: 1 ：Step 1: First, remember that the difference quotient is defined as [frac{f(x + h) - f(x)}{h}]Step: 2 ：Step 2: Substitute (f(x + h)) and (f(x)) into the formula. [frac{(3(x + h)^2 + 2(x + h) + 1) - (3x^2 + 2x + 1)}{h}]Step: 3 ：Step 3: Expand and simplify the numerator.[frac{(3x^2 + 6xh + 3h^2 + 2x + 2h + 1) - (3x^2 + 2x + 1)}{h}] [=frac{6xh + 3h^2 + 2h}{h}]Step: 4 ：Step 4: Cancel out (h) from the numerator and denominator. [= 6x + 3h + 2]

Evaluate the difference quotient for the function \(f(x) = 3x^2 + 2x + 1\).

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