Problem

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

Solution

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. \[= 6x + 3h + 2\]

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Source: https://solvelyapp.com/problems/onoEc1fHyF/

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