Find d/dx(ln(x/x²+1)) without using the Quotient Rule.

Verified step by step guidance

1. First, recognize that the function inside the logarithm is a composition of functions: ln(u), where u = x/(x² + 1).
2. Apply the chain rule for differentiation: d/dx[ln(u)] = (1/u) * du/dx.
3. To find du/dx, express u = x/(x² + 1) as a product: u = x * (x² + 1)^(-1).
4. Use the product rule to differentiate u: d/dx[x * (x² + 1)^(-1)] = (d/dx[x]) * (x² + 1)^(-1) + x * d/dx[(x² + 1)^(-1)].
5. Differentiate (x² + 1)^(-1) using the chain rule: d/dx[(x² + 1)^(-1)] = -1 * (x² + 1)^(-2) * d/dx[x² + 1].

Key Concepts

Natural Logarithm Properties

The natural logarithm, denoted as ln, has specific properties that simplify expressions. One key property is that ln(a/b) = ln(a) - ln(b). This property allows us to break down complex logarithmic expressions into simpler components, making differentiation easier without applying the Quotient Rule.

Chain Rule

The Chain Rule is a fundamental differentiation technique used when dealing with composite functions. It states that if a function y = f(g(x)), then the derivative dy/dx = f'(g(x)) * g'(x). This rule is essential for differentiating functions where one function is nested within another, which is often the case in logarithmic expressions.

Simplifying Expressions Before Differentiation

Before differentiating, simplifying expressions can significantly ease the process. This involves rewriting complex functions in a more manageable form, such as factoring or using logarithmic identities. By simplifying ln(x/(x²+1)) to ln(x) - ln(x²+1), we can apply the properties of logarithms to differentiate each term separately, avoiding the need for the Quotient Rule.