Find the slope of the curve y=sin^-1 x at (1/2, π/6) without calculating the derivative of sin^-1 x.

검증된 단계별 안내

Step 1: Recognize that the slope of the curve at a given point is the derivative of the function at that point. Here, we need the derivative of \( y = \sin^{-1}(x) \).

Step 2: Use the identity \( y = \sin^{-1}(x) \) implies \( \sin(y) = x \). Differentiate both sides with respect to \( x \).

Step 3: Apply implicit differentiation. The derivative of \( \sin(y) \) with respect to \( x \) is \( \cos(y) \cdot \frac{dy}{dx} \), and the derivative of \( x \) is 1.

Step 4: Set up the equation from implicit differentiation: \( \cos(y) \cdot \frac{dy}{dx} = 1 \). Solve for \( \frac{dy}{dx} \) to find \( \frac{dy}{dx} = \frac{1}{\cos(y)} \).

Step 5: Evaluate \( \cos(y) \) at the point \( (1/2, \pi/6) \). Since \( y = \pi/6 \), \( \cos(\pi/6) = \sqrt{3}/2 \). Substitute this into the expression for \( \frac{dy}{dx} \) to find the slope.

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주요 개념

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

Inverse Functions

Inverse functions reverse the effect of the original function. For example, if y = sin(x), then x = sin<sup>-1</sup>(y) is the inverse function. Understanding how to work with inverse functions is crucial for analyzing their properties, such as slopes and behavior at specific points.

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the function at that point. It can be interpreted as the tangent line's steepness at the point of interest. For the curve y = sin<sup>-1</sup>(x), finding the slope at (1/2, π/6) involves understanding the relationship between the function and its inverse.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this context, knowing the values of sin and cos at specific angles, such as π/6, is essential for determining the slope without directly calculating the derivative. These identities help relate the angles to their sine and cosine values.

Find the slope of the curve y=sin-1 x at (1/2, π/6) without calculating the derivative of sin-1 x.