Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/3

Verified step by step guidance

1. First, understand that the problem requires finding the equation of the tangent line to the curve y = 4 sin x cos x at x = π/3. The tangent line will have the form y = mx + b, where m is the slope at the given point and b is the y-intercept.
2. To find the slope of the tangent line, we need to compute the derivative of the function y = 4 sin x cos x with respect to x. Use the product rule for differentiation, which states that if you have a function u(x)v(x), its derivative is u'(x)v(x) + u(x)v'(x).
3. Apply the product rule: Let u(x) = 4 sin x and v(x) = cos x. Then, u'(x) = 4 cos x and v'(x) = -sin x. The derivative of y with respect to x, denoted as dy/dx, is: dy/dx = u'(x)v(x) + u(x)v'(x) = 4 cos x * cos x + 4 sin x * (-sin x).
4. Simplify the expression for dy/dx: dy/dx = 4 cos^2 x - 4 sin^2 x. This can be further simplified using the trigonometric identity cos(2x) = cos^2 x - sin^2 x, giving dy/dx = 4 cos(2x).
5. Evaluate the derivative at x = π/3 to find the slope m of the tangent line: m = 4 cos(2 * π/3). Calculate the value of y at x = π/3 using the original function y = 4 sin x cos x, and use these values to find the equation of the tangent line in the form y = mx + b.

Key Concepts

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 is equal to the derivative of the function at that point. This concept is crucial for finding the equation of the tangent line, as it provides both the slope and the point of tangency.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, calculating the derivative of the function y = 4 sin x cos x will allow us to find the slope of the tangent line at x = π/3.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. In this problem, the identity sin(2x) = 2 sin x cos x can simplify the function y = 4 sin x cos x to y = 2 sin(2x), making it easier to differentiate and evaluate at specific points.