f'(u) = \frac{(1 + 9u)^2 \cdot 1 - u \cdot 2(1 + 9u)(9)}{(1 + 9u)^4}.

["# Simplify and Analyze the Derivative: ( f'(u) = \frac{(1 + 9u)^2 \cdot 1 - u \cdot 2(1 + 9u)(9)}{(1 + 9u)^4} )", "Understanding derivatives is essential in calculus, especially when analyzing function behavior—such as rates of change, extrema, and curves’ slopes. Today, we break down the derivative expression:", "[\nf'(u) = \frac{(1 + 9u)^2 - 18u(1 + 9u)}{(1 + 9u)^4}\n]", "This form results from carefully expanding and simplifying the original expression. Let’s explore its meaning and how to simplify and interpret it.", "---", "## Step 1: Simplify the Numerator", "Start with the given numerator:", "[\n(1 + 9u)^2 - 18u(1 + 9u)\n]", "Factor out the common term ( (1 + 9u) ):", "[\n(1 + 9u)\left[(1 + 9u) - 18u\right]\n]", "Simplify inside the brackets:", "[\n(1 + 9u)(1 + 9u - 18u) = (1 + 9u)(1 - 9u)\n]", "Notice this is a difference of squares:", "[\n(1 + 9u)(1 - 9u) = 1^2 - (9u)^2 = 1 - 81u^2\n]", "So the numerator simplifies to ( 1 - 81u^2 ), and the full derivative becomes:", "[\nf'(u) = \frac{1 - 81u^2}{(1 + 9u)^4}\n]", "---", "## Step 2: Final Form and Observations", "The simplified form of the derivative is:", "[\nf'(u) = \frac{1 - 81u^2}{(1 + 9u)^4}\n]", "### Key characteristics:", "- Domain: The function is defined for all real ( u ) such that ( 1 + 9u <br/>\ne 0 ), i.e., ( u <br/>\ne -\frac{1}{9} ).", "- Critical Points: Set numerator ( f'(u) = 0 ) to find horizontal tangents:", "[\n1 - 81u^2 = 0 \Rightarrow u^2 = \frac{1}{81} \Rightarrow u = \pm \frac{1}{9}\n]", "But since ( u = -\frac{1}{9} ) makes the denominator zero (undefined), only ( u = \frac{1}{9} ) is a valid critical point.", "- Behavior near ( u = -\frac{1}{9} ): The denominator approaches zero while the numerator remains non-zero, so the derivative approaches ( \pm\infty ), indicating a vertical asymptote and possible cusp or discontinuity.", "- Sign Analysis: The sign of ( f'(u) ) depends on the numerator ( 1 - 81u^2 ):", "- Positive when ( |u| < \frac{1}{9} )\n - Negative when ( |u| > \frac{1}{9} )", "This suggests ( f(u) ) increases on ( \left(-\infty, -\frac{1}{9}\right) ) and ( \left(-\frac{1}{9}, \frac{1}{9}\right) ), then decreases for ( |u| > \frac{1}{9} ), suggesting a local maximum near ( u = \frac{1}{9} ).", "---", "## Step 3: Why This Derivative Matters", "Analyzing this derivative helps in sketching ( f(u) ), locating extrema, and understanding function concavity and continuity. Despite being algebraically complex, simplifying ( f'(u) ) reveals important features that guide graphing and application in optimization problems.", "---", "## SEO Keywords to Boost Visibility", "- Derivative simplification\n- Calculus: ( f'(u) ) analysis\n- Simplify ( \frac{(1 + 9u)^2 - 18u(1 + 9u)}{(1 + 9u)^4} )\n- Calculus derivative example\n- Critical points and function behavior\n- Derivative interpretation and graphing", "---", "## Conclusion", "While the expression\n[\nf'(u) = \frac{(1 + 9u)^2 \cdot 1 - u \cdot 2(1 + 9u)(9)}{(1 + 9u)^4}\n]\nmay appear daunting, breaking it down shows key insights into its simplified form and the behavior of the original function. Mastering such derivatives enhances proficiency in calculus, empowering deeper mathematical analysis and solution-solving.", "For further study, consider graphing this function or exploring its integrals to understand accumulation—critical steps in advanced calculus applications.", "---", "Ready to simplify more derivatives or explore related calculus tools? Drop a comment or connect for more explanatory guides!"]