Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In 2x/(x² + 1)³

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First, recognize that the function f(x) = ln(2x/(x² + 1)³) is a composition of functions, specifically a logarithmic function applied to a rational function. To differentiate it, we can use the chain rule.

Apply the properties of logarithms to simplify the function: ln(2x/(x² + 1)³) can be rewritten using the property ln(a/b) = ln(a) - ln(b). Thus, f(x) = ln(2x) - ln((x² + 1)³).

Further simplify using the property ln(a^b) = b*ln(a). Therefore, f(x) = ln(2x) - 3*ln(x² + 1).

Differentiate each term separately. For ln(2x), use the chain rule: the derivative of ln(u) is 1/u * du/dx. Here, u = 2x, so the derivative is 1/(2x) * 2 = 1/x.

For the term -3*ln(x² + 1), again use the chain rule: the derivative of ln(v) is 1/v * dv/dx. Here, v = x² + 1, so the derivative is -3 * (1/(x² + 1) * 2x) = -6x/(x² + 1). Combine these results to find f'(x).

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Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the function's graph at any given point. The derivative is denoted as f'(x) and can be calculated using various rules, such as the power rule, product rule, and quotient rule.

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Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). These properties are particularly useful in calculus for simplifying complex functions before differentiation.

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Quotient Rule

The quotient rule is a formula used to find the derivative of a function that is the ratio of two other functions. If f(x) = g(x)/h(x), the derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))². This rule is essential when differentiating functions that are expressed as fractions, allowing for accurate computation of their rates of change.

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