Find the derivative of \( f(x) = 3x^2 + 5x - 7 \).
["# Find the Derivative of ( f(x) = 3x^2 + 5x - 7 )", "Understanding calculus is essential in mathematics, particularly when working with functions in algebra, physics, and engineering. One of the most fundamental operations is finding the derivative of a function. In this article, we’ll explore how to find the derivative of the function:", "[\nf(x) = 3x^2 + 5x - 7\n]", "## What Is a Derivative?", "The derivative of a function represents the rate at which the function’s value changes with respect to a change in the variable ( x ). In practical terms, it gives the slope of the tangent line to the function’s graph at any point.", "For polynomial functions like ( f(x) ), we use coefficient rules and the power rule to compute derivatives efficiently.", "## Applying the Power Rule", "To differentiate ( f(x) = 3x^2 + 5x - 7 ), we apply the power rule, which states that:", "[\n\frac{d}{dx}[x^n] = nx^{n-1}\n]", "We’ll also use the constant multiple rule and sum/difference rules, which say derivatives of sums/differences of functions are the sums/differences of their derivatives.", "Let’s break down each term:", "### 1. Differentiating ( 3x^2 )", "Using the power rule:", "[\n\frac{d}{dx}[3x^2] = 3 \cdot \frac{d}{dx}[x^2] = 3 \cdot 2x^{2-1} = 6x\n]", "### 2. Differentiating ( 5x )", "Treating ( 5x ) as ( 5x^1 ):", "[\n\frac{d}{dx}[5x] = 5 \cdot \frac{d}{dx}[x^1] = 5 \cdot 1x^{1-1} = 5 \cdot 1 = 5\n]", "### 3. Differentiating the constant ( -7 )", "The derivative of any constant is zero:", "[\n\frac{d}{dx}[-7] = 0\n]", "## Putting It All Together", "Now combine the derivatives of each term:", "[\nf'(x) = \frac{d}{dx}[3x^2 + 5x - 7] = 6x + 5 + 0 = 6x + 5\n]", "## Final Answer", "[\n\boxed{f'(x) = 6x + 5}\n]", "## Why This Derivative Matters", "The derivative ( f'(x) = 6x + 5 ) tells us the instantaneous rate of change of ( f(x) ) at any point ( x ). This information is valuable in:", "- Finding critical points for optimization problems\n- Analyzing the slope of a curve at a specific location\n- Modeling motion in physics, where ( f(x) ) represents position and ( f'(x) ) is velocity", "By mastering derivatives of basic polynomials like ( f(x) = 3x^2 + 5x - 7 ), you’re building a strong foundation for more complex calculus topics such as integration, higher-degree polynomials, and multivariable functions.", "---", "If you’re studying calculus, practice differentiating various polynomials to become fluent with the power rule and related techniques. For more detailed guides and step-by-step examples, explore online calculus aides or consult your textbook!"]
