Calculate the derivative of \(f(x) = 3x^4 - 5x^2 + 7x - 9\) with respect to \(x\).

["Calculate the Derivative of (f(x) = 3x^4 - 5x^2 + 7x - 9) with Respect to (x)", "Finding derivatives is a fundamental skill in calculus, and understanding how to compute the derivative of a function helps in areas like optimization, motion analysis, and curve sketching. Today, we’ll walk through the step-by-step process of calculating the derivative of the polynomial function:", "[\nf(x) = 3x^4 - 5x^2 + 7x - 9\n]", "### What is a Derivative?", "The derivative of a function (f(x)) with respect to (x), written as (f'(x)) or (\frac{df}{dx}), represents the rate of change of the function at any point (x). For polynomial functions, differentiation follows simple power rule rules.", "---", "### Step-by-Step Derivative Calculation", "Let’s apply the Power Rule of differentiation, which states:", "> If (f(x) = ax^n), then (f'(x) = a \cdot n \cdot x^{n-1})", "We’ll differentiate each term in (f(x)) individually.", "#### 1. Differentiate (3x^4)", "Using the power rule:\n[\n\frac{d}{dx}(3x^4) = 3 \cdot 4x^{4-1} = 12x^3\n]", "#### 2. Differentiate (-5x^2)", "Again applying the power rule:\n[\n\frac{d}{dx}(-5x^2) = -5 \cdot 2x^{2-1} = -10x\n]", "#### 3. Differentiate (7x)", "Since (x) is (x^1), we apply the rule:\n[\n\frac{d}{dx}(7x) = 7 \cdot 1x^{1-1} = 7x^0 = 7\n]", "#### 4. Differentiate (-9)", "The derivative of a constant is always zero:\n[\n\frac{d}{dx}(-9) = 0\n]", "---", "### Combine All Terms", "Now, add the derivatives of each term:", "[\nf'(x) = 12x^3 - 10x + 7 + 0\n]", "### Final Result", "[\n\boxed{f'(x) = 12x^3 - 10x + 7}\n]", "---", "### Why This Matters", "This derivative helps analyze the function’s behavior:", "- Slope: (f'(x)) gives the slope of the tangent line to (f(x)) at any point (x).\n- Critical Points: Setting (f'(x) = 0) identifies potential maxima, minima, or inflection points.\n- Growth Rate: The leading term (12x^3) indicates how the function grows as (x) increases.", "Mastering basic differentiation like this equips you with foundational calculus tools essential for science, engineering, economics, and beyond.", "---", "Keywords: derivative, calculus, differentiate, (f(x) = 3x^4 - 5x^2 + 7x - 9), power rule, (f'(x)), math tutorial, solve derivative, slope, rate of change."]