75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = tan¹⁰x / (5x+3)⁶

검증된 단계별 안내

Step 1: Begin by taking the natural logarithm of both sides of the equation y = (tan^10(x)) / (5x + 3)^6. This gives ln(y) = ln((tan^10(x)) / (5x + 3)^6).

Step 2: Use the properties of logarithms to simplify the expression. The logarithm of a quotient is the difference of the logarithms: ln(y) = ln(tan^10(x)) - ln((5x + 3)^6).

Step 3: Apply the power rule for logarithms, which states that ln(a^b) = b*ln(a). This gives ln(y) = 10*ln(tan(x)) - 6*ln(5x + 3).

Step 4: Differentiate both sides with respect to x. The left side becomes (1/y) * dy/dx by implicit differentiation. The right side requires the chain rule: 10 * (1/tan(x)) * sec^2(x) - 6 * (1/(5x + 3)) * 5.

Step 5: Solve for dy/dx by multiplying both sides by y. Substitute back y = (tan^10(x)) / (5x + 3)^6 to express dy/dx in terms of x. This gives dy/dx = y * [10 * (1/tan(x)) * sec^2(x) - 6 * (1/(5x + 3)) * 5].

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 functions that are products or quotients of other functions, especially when they involve powers. By taking the natural logarithm of both sides of the function, we can simplify the differentiation process, allowing us to use properties of logarithms to break down complex expressions into manageable parts.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential when dealing with functions raised to powers or nested functions.

Product and Quotient Rules

The product and quotient rules are essential for differentiating functions that are products or ratios of two or more functions. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. The quotient rule, on the other hand, provides a method for differentiating a ratio of two functions, ensuring accurate results when applying these rules in logarithmic differentiation.

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).f(x) = tan¹⁰x / (5x+3)⁶