Evaluate and simplify y'.

y = ln |sec 3x|

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Identify the given equation: y' * y = ln |sec(3x)|. This is a differential equation where y' is the derivative of y with respect to x.

Recognize that this is a separable differential equation. We can separate the variables y and x to solve it.

Rewrite the equation by dividing both sides by y: y' = (ln |sec(3x)|) / y.

Integrate both sides: Integrate the left side with respect to y and the right side with respect to x. The left side becomes ∫(1/y) dy, and the right side becomes ∫ln |sec(3x)| dx.

Solve the integrals: The integral of 1/y with respect to y is ln|y|, and the integral of ln |sec(3x)| with respect to x will require integration techniques such as integration by parts. After integrating, solve for y to find the general solution.

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주요 개념

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

Derivative of Natural Logarithm

The derivative of the natural logarithm function, ln(x), is given by 1/x. When dealing with the natural logarithm of a function, such as ln|u|, the derivative is found using the chain rule, resulting in (1/u) * (du/dx). This principle is essential for differentiating functions involving logarithms.

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 another function u, then the derivative dy/dx can be calculated as dy/du multiplied by du/dx. This rule is crucial when differentiating functions like ln|sec 3x|, where sec 3x is a composite function.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is vital for calculus. For example, the derivative of sec(x) is sec(x)tan(x). In the context of the given function, knowing how to differentiate sec(3x) will be necessary to apply the chain rule effectively and find the derivative of y = ln|sec 3x|.

Evaluate and simplify y'.y = ln |sec 3x|