Find the derivative of the function \( f(x) = 3x^2 + 2x - 5 \).
["# Find the Derivative of the Function ( f(x) = 3x^2 + 2x - 5 )", "Understanding derivatives is essential in calculus because they represent the rate of change of a function at any given point. In this article, we’ll walk through how to find the derivative of the function ( f(x) = 3x^2 + 2x - 5 ) step by step, including a clear explanation of the key rules used in differentiation.", "## What is a Derivative?", "The derivative of a function ( f(x) ), denoted ( f'(x) ) or ( \frac{df}{dx} ), measures how the function’s output changes as its input ( x ) changes. For polynomial functions, differentiation follows a set of standard rules that make the process systematic and straightforward.", "## The Function to Differentiate", "We are given the quadratic function:", "[\nf(x) = 3x^2 + 2x - 5\n]", "This function is a sum of terms: a quadratic term ( 3x^2 ), a linear term ( 2x ), and a constant ( -5 ).", "---", "## Step-by-Step Derivative Calculation", "Each term in the function is a power of ( x ) multiplied by a coefficient. We use the Power Rule of differentiation, which states:", "[\n\frac{d}{dx}[x^n] = nx^{n-1}\n]", "Additionally, the derivative of a constant is zero, and derivatives operate linearly, allowing us to differentiate term by term.", "### 1. Differentiate ( 3x^2 )", "Apply the power rule:", "[\n\frac{d}{dx}[3x^2] = 3 \cdot 2x^{2-1} = 6x\n]", "### 2. Differentiate ( 2x )", "This is critical: the derivative of ( x ) is 1, and multiplying by 2 remains:", "[\n\frac{d}{dx}[2x] = 2 \cdot 1 = 2\n]", "### 3. Differentiate ( -5 )", "Since (-5) is a constant:", "[\n\frac{d}{dx}[-5] = 0\n]", "---", "## Combine the Results", "Now, combine the derivatives of all terms:", "[\nf'(x) = 6x + 2 + 0 = 6x + 2\n]", "---", "## Final Answer", "The derivative of the function ( f(x) = 3x^2 + 2x - 5 ) is:", "[\n\boxed{f'(x) = 6x + 2}\n]", "---", "## Why This Matters", "Calculating derivatives helps solve real-world problems involving rates of change, optimization, and modeling. For example, if ( f(x) ) represents the position of a moving object over time, its derivative ( f'(x) ) represents instantaneous velocity.", "Understanding how to differentiate polynomial functions like ( f(x) = 3x^2 + 2x - 5 ) lays the foundation for more complex applications in physics, economics, and engineering.", "---", "## Search Terms This Article Covers", "- Find the derivative of ( 3x^2 + 2x - 5 )\n- Differentiate quadratic function\n- Power rule application\n- Derivative step-by-step\n- Calculus derivatives for beginners", "Master these concepts using this guide and improve your calculus skills today!"]
