Find the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \) and evaluate it at \( x = 2 \).
["# Finding the Derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) and Evaluating It at ( x = 2 )", "Understanding how to compute derivatives is fundamental in calculus, especially for analyzing functions in science, engineering, and economics. In this article, we’ll walk through finding the derivative of the function ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), and evaluate it at ( x = 2 ) step by step.", "## What Is a Derivative?", "The derivative of a function represents the rate of change of that function with respect to its variable. For a polynomial function like ( f(x) ), the derivative can be found using basic differentiation rules such as the power rule.", "## Step-by-Step: Differentiate ( f(x) = 3x^3 - 5x^2 + 2x - 7 )", "Using the power rule—where ( \frac{d}{dx}[x^n] = nx^{n-1} )—we differentiate each term individually:", "1. Differentiate ( 3x^3 ):\n ( \frac{d}{dx}[3x^3] = 3 \cdot 3x^{3-1} = 9x^2 )", "2. Differentiate ( -5x^2 ):\n ( \frac{d}{dx}[-5x^2] = -5 \cdot 2x^{2-1} = -10x )", "3. Differentiate ( 2x ):\n ( \frac{d}{dx}[2x] = 2 \cdot 1x^{1-1} = 2 )", "4. Differentiate the constant term ( -7 ):\n ( \frac{d}{dx}[-7] = 0 )", "Combining all these results, the derivative of ( f(x) ) is:", "[\nf'(x) = 9x^2 - 10x + 2\n]", "## Evaluating the Derivative at ( x = 2 )", "Now that we have ( f'(x) = 9x^2 - 10x + 2 ), substitute ( x = 2 ) into the expression:", "[\nf'(2) = 9(2)^2 - 10(2) + 2\n]", "Calculate step by step:", "- ( 2^2 = 4 )\n- ( 9 \cdot 4 = 36 )\n- ( 10 \cdot 2 = 20 )", "So,", "[\nf'(2) = 36 - 20 + 2 = 18\n]", "## Conclusion", "The derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) is:", "[\n\boxed{f'(x) = 9x^2 - 10x + 2}\n]", "Evaluating at ( x = 2 ), the derivative is:", "[\n\boxed{f'(2) = 18}\n]", "This means the function’s instantaneous rate of change at ( x = 2 ) is 18, which is valuable for optimization and curve analysis.", "---", "### Why This Matters", "Knowing derivatives helps solve real-world problems such as finding maximum profits, rate of reaction in chemistry, or velocity from position data. Mastering this skill enhances your ability to work with mathematical models across disciplines.", "For further practice, try differentiating other polynomial expressions or apply the derivative to model physical phenomena!"]
