Let f(x) = sin x. What is the value of f′(π)?

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First, identify the function f(x) = sin(x). We need to find the derivative of this function, which is f′(x).

Recall the derivative rule for the sine function: the derivative of sin(x) with respect to x is cos(x). Therefore, f′(x) = cos(x).

Now, substitute x = π into the derivative function f′(x) = cos(x) to find f′(π).

Evaluate cos(π). Remember that the cosine of π radians is a known trigonometric value.

Conclude by stating the value of f′(π) based on the evaluation of cos(π).

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주요 개념

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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 defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, we need to find the derivative of the sine function, which will help us determine the slope of the tangent line at a specific point.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to side lengths in right triangles. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding the properties and values of these functions at key angles (like π) is essential for evaluating derivatives and solving related problems.

Value of f′(x) at Specific Points

To find the value of the derivative at a specific point, we evaluate the derivative function at that point. For the sine function, we first compute its derivative, which is cosine. Then, we substitute the specific angle (in this case, π) into the derivative to find the slope of the tangent line at that point, which provides the answer to the original question.