Differentiate term by term: \(f'(x) = 9x^2 - 10x + 6\).

["# Differentiating (f'(x) = 9x^2 - 10x + 6): A Clear Term-by-Term Breakdown", "Calculus is a powerful branch of mathematics that helps us understand how functions change — and one of its core tools is differentiation. Understanding the derivative (f'(x) = 9x^2 - 10x + 6) requires breaking it down term by term, identifying each component, and explaining how they behave. This article guides you through differentiating this quadratic polynomial step by step, so you can master not just the result, but the method itself.", "---", "## What is a Derivative – A Quick Refresher", "The derivative (f'(x)) represents the instantaneous rate of change or slope of a function (f(x)) at any point (x). For polynomials, differentiation follows straightforward rules: the derivative of (x^n) is (nx^{n-1}), constants disappear, and powers are reduced systematically. In this case, (f'(x)) is already expressed in expanded form, making it ready for term-by-term analysis.", "---", "## Step 1: Recognize the Structure of (f'(x))", "We are given:\n[\nf'(x) = 9x^2 - 10x + 6\n]", "This is a quadratic polynomial composed of three distinct terms, each involving a power of (x) multiplied by a coefficient. Structurally, it can be written as:\n[\nf'(x) = (\ ext{Term 1}) + (\ ext{Term 2}) + (\ ext{Term 3}) = 9x^2 + (-10x) + 6\n]", "---", "## Step 2: Apply the Power Rule to Each Term", "The power rule is fundamental:\n[\n\frac{d}{dx}[x^n] = nx^{n-1}\n]", "Apply this rule to each term individually.", "### Term 1: (9x^2)\n- Coefficient: 9\n- Power: 2\n- Apply the power rule:\n[\n\frac{d}{dx}[9x^2] = 9 \cdot 2x^{2-1} = 18x\n]", "### Term 2: (-10x)\n- Rewrite (x) as (x^1):\n- Coefficient: (-10)\n- Power: 1\n- Apply the power rule:\n[\n\frac{d}{dx}[-10x] = -10 \cdot 1x^{1-1} = -10 \cdot 1x^0 = -10 \cdot 1 = -10\n]\n(Terminally, (x^0 = 1), so the derivative is (-10), a constant.)", "### Term 3: (6)\n- This is a constant.\n- The derivative of any constant is zero:\n[\n\frac{d}{dx}[6] = 0\n]", "---", "## Step 3: Combine the Derivatives", "Now sum the derivatives of each term:\n[\nf'(x) = \underbrace{18x}{\ ext{from } 9x^2} + \underbrace{(-10)} = 18x - 10} -10x} + \underbrace{0}_{\ ext{from } 6\n]", "---", "## Final Result", "Thus, the derivative of (f(x)) is:\n[\nf'(x) = 9x^2 - 10x + 6 \quad \Rightarrow \quad f'(x) = 18x - 10 \quad \ ext{(after differentiation)}\n]", "---", "## Why Differentiating Term by Term Matters", "Breaking down (f'(x)) term by term allows us to:\n- Identify how each power of (x) contributes to the slope,\n- Verify the derivative using fundamental rules,\n- Apply the same method to more complex functions—polynomials, trigonometric expressions, or product/quotient terms.", "Understanding this approach builds a strong foundation for calculus and its applications in physics, engineering, economics, and data science.", "---", "## Summary Table: Term-by-Term Breakdown", "| Original Term | Form | Power Rule Application | Derivative | Result |\n|--------------------------|---------------|-------------------------------|------------------|-----------------|\n| (9x^2) | Coefficient × (x^2) | (2 \cdot 9x^{2-1} = 18x) | (18x) | (18x) |\n| (-10x) | (-10x^1) | (-10 \cdot 1x^{0} = -10) | (-10) | (-10) |\n| (6) | Constant | (0) (no (x)) | (0) | (0) |", "---", "## Conclusion", "Differentiating (f'(x) = 9x^2 - 10x + 6) term by term reveals how each component shapes the function’s rate of change. By practicing this structured approach—identifying terms, applying the power rule, and summing results—you develop both precision and intuition in calculus. Whether you’re solving for (f(x)) or analyzing motion and growth, mastering term-by-term differentiation is essential. Keep practicing, and soon differentiation will become second nature!", "---", "Keywords: differentiation, (f'(x) = 9x^2 - 10x + 6), term-by-term differentiation, power rule, calculus fundamentals, rate of change, algebraic differentiation, polynomial derivative.\nMeta Description: Learn how to differentiate (f'(x) = 9x^2 - 10x + 6) by term. Step-by-step breakdown using the power rule, ideal for students mastering calculus."]