To find the derivative \( f'(x) \), differentiate the function \( f(x) = 3x^2 - 2x + 1 \).
["# How to Find the Derivative ( f'(x) ) of ( f(x) = 3x^2 - 2x + 1 )", "Learning calculus is essential for understanding how functions change, and one of the core concepts is finding derivatives. If you’re working with the function:", "[\nf(x) = 3x^2 - 2x + 1\n]", "you’re about to discover how to compute its derivative ( f'(x) ) using basic differentiation rules.", "## What is a Derivative?", "The derivative ( f'(x) ) represents the rate at which the function ( f(x) ) changes at any point ( x ). It’s the foundation of calculus and is widely used in physics, engineering, economics, and data science.", "## Differentiating Each Term", "Let’s apply fundamental differentiation rules step by step.", "### Step 1: Differentiate ( 3x^2 )", "Use the power rule, which states:\n[\n\frac{d}{dx}[x^n] = nx^{n-1}\n]", "For ( 3x^2 ):\n- The coefficient 3 remains.\n- Apply the power rule:\n[\n\frac{d}{dx}[3x^2] = 3 \cdot 2x^{2-1} = 6x\n]", "### Step 2: Differentiate ( -2x )", "The term ( -2x ) can be written as ( -2x^1 ).\nAgain using the power rule:\n[\n\frac{d}{dx}[-2x] = -2 \cdot 1x^{1-1} = -2x^0 = -2\n]", "### Step 3: Differentiate the constant ( +1 )", "The derivative of any constant is zero:\n[\n\frac{d}{dx}[1] = 0\n]", "## Putting It All Together", "Now combine the derivatives of each term:", "[\nf'(x) = \frac{d}{dx}[3x^2] + \frac{d}{dx}[-2x] + \frac{d}{dx}[1] = 6x - 2 + 0\n]", "Thus, the derivative is:", "[\nf'(x) = 6x - 2\n]", "## Why This Matters", "Finding derivatives helps you analyze function behavior — for instance, identifying where the function increases or decreases, locating peaks and valleys, or determining optimization points.", "This simple quadratic function illustrates core differentiation principles that extend to more complex functions.", "## Summary", "- Function: ( f(x) = 3x^2 - 2x + 1 )\n- Derivative: ( f'(x) = 6x - 2 )\n- Key tools: Power rule, constant rule", "Mastering this step-by-step approach builds confidence for tackling higher-level calculus concepts.", "---", "Keywords: derivative, ( f'(x) ), differentiation, calculus tutorial, power rule, find derivative, derivative of ( 3x^2 - 2x + 1 ), math help, derivative rules.", "For more practice, try differentiating polynomial functions — they’re ideal for applying foundational calculus techniques."]
