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 - 1)(x + 2)³ / (1 - 4x)²

검증된 단계별 안내

First, recognize that the function f(x) = ln((2x - 1)(x + 2)^3 / (1 - 4x)^2) is a natural logarithm of a quotient. Use the logarithmic property ln(a/b) = ln(a) - ln(b) to separate the terms.

Apply the property of logarithms ln(a * b) = ln(a) + ln(b) to further break down the expression: ln((2x - 1)(x + 2)^3) - ln((1 - 4x)^2).

Use the power rule for logarithms, ln(a^b) = b * ln(a), to simplify the expression: ln(2x - 1) + 3 * ln(x + 2) - 2 * ln(1 - 4x).

Differentiate each term separately using the chain rule. For ln(2x - 1), the derivative is 1/(2x - 1) * 2. For 3 * ln(x + 2), the derivative is 3 * (1/(x + 2)). For -2 * ln(1 - 4x), the derivative is -2 * (-4/(1 - 4x)).

Combine the derivatives from each term to find f'(x): f'(x) = 2/(2x - 1) + 3/(x + 2) + 8/(1 - 4x).

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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 can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.

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.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and x (i.e., y = f(u) and u = g(x)), then the derivative of y with respect to x is the product of the derivative of f with respect to u and the derivative of g with respect to x. This rule is essential for handling functions that are nested within each other.

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 - 1)(x + 2)³ / (1 - 4x)²