In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))

검증된 단계별 안내

Start by rewriting the function to make logarithmic differentiation easier. Let \( y = \sqrt[3]{\frac{x(x+1)(x-2)}{(x^2+1)(2x+3)}} \). This can be expressed as \( y = \left( \frac{x(x+1)(x-2)}{(x^2+1)(2x+3)} \right)^{\frac{1}{3}} \).

Take the natural logarithm of both sides to simplify the differentiation process: \( \ln y = \frac{1}{3} \ln \left( \frac{x(x+1)(x-2)}{(x^2+1)(2x+3)} \right) \).

Use the logarithm property for quotients and products to expand the right side: \( \ln y = \frac{1}{3} \left[ \ln x + \ln (x+1) + \ln (x-2) - \ln (x^2+1) - \ln (2x+3) \right] \).

Differentiate both sides with respect to \( x \). Remember that \( \frac{d}{dx} (\ln y) = \frac{1}{y} \frac{dy}{dx} \) by implicit differentiation. So, \( \frac{1}{y} \frac{dy}{dx} = \frac{1}{3} \left( \frac{1}{x} + \frac{1}{x+1} + \frac{1}{x-2} - \frac{2x}{x^2+1} - \frac{2}{2x+3} \right) \).

Finally, solve for \( \frac{dy}{dx} \) by multiplying both sides by \( y \), and then substitute back the original expression for \( y \) to express the derivative in terms of \( x \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate complex functions, especially those involving products, quotients, or powers. By taking the natural logarithm of both sides, the function is simplified into sums and differences, making differentiation easier. This method is particularly useful when the function is a product or quotient of multiple expressions.

Properties of Logarithms

The properties of logarithms, such as log(ab) = log a + log b and log(a/b) = log a - log b, allow us to break down complicated expressions into simpler parts. These rules help transform products into sums and quotients into differences, which simplifies the differentiation process after taking the logarithm of the function.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. After applying logarithmic differentiation, the resulting expression often involves functions inside other functions, requiring the chain rule to differentiate correctly. It states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))