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

Guida verificata passo dopo passo

First, recognize that the function inside the logarithm is a composition of functions: ln(u), where u = x/(x² + 1).

Apply the chain rule for differentiation: d/dx[ln(u)] = (1/u) * du/dx.

To find du/dx, express u = x/(x² + 1) as a product: u = x * (x² + 1)^(-1).

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)].

Differentiate (x² + 1)^(-1) using the chain rule: d/dx[(x² + 1)^(-1)] = -1 * (x² + 1)^(-2) * d/dx[x² + 1].

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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.

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Change of Base Property

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.

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Intro to the Chain Rule

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.

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