{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.

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First, find the derivative of the function y = 2^sin(x) with respect to x. Use the chain rule and the fact that the derivative of a^u, where a is a constant, is a^u * ln(a) * du/dx.

Evaluate the derivative at x = π/2. Substitute x = π/2 into the derivative to find the slope of the tangent line at this point.

Calculate the value of the function y = 2^sin(x) at x = π/2 to find the y-coordinate of the point of tangency.

Use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope found in step 2, and (x1, y1) is the point of tangency found in step 3, to write the equation of the tangent line.

Graph the function y = 2^sin(x) and the tangent line using a graphing tool or software. Ensure that the tangent line touches the curve at the point (π/2, 2^sin(π/2)).

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주요 개념

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

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative. In this case, we need to find the derivative of the function y=2^sin(x) at x=π/2 to determine the slope of the tangent line.

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. For the function y=2^sin(x), we will apply the chain rule to differentiate it, which is essential for finding the slope of the tangent line at the specified point.

Exponential Functions

Exponential functions are mathematical functions of the form y=a^x, where 'a' is a positive constant. In this case, y=2^sin(x) is an exponential function where the exponent is a trigonometric function. Understanding the behavior of exponential functions, especially how they change with respect to their exponents, is crucial for analyzing the function and its tangent line.