Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

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First, identify the given functions: y = 3 sin(2x) and x = t² + π. We need to find dy/dt at t = 0.

Use the chain rule to find dy/dt. The chain rule states that dy/dt = (dy/dx) * (dx/dt).

Calculate dy/dx. Since y = 3 sin(2x), use the derivative of sin(u), which is cos(u) * du/dx. Here, u = 2x, so dy/dx = 3 * cos(2x) * 2 = 6 cos(2x).

Calculate dx/dt. Since x = t² + π, differentiate with respect to t to get dx/dt = 2t.

Substitute dy/dx and dx/dt into the chain rule expression: dy/dt = 6 cos(2x) * 2t. Evaluate this expression at t = 0, where x = (0)² + π = π, to find the value of dy/dt.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a variable y depends on u, and u depends on t, then the derivative of y with respect to t can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to t. This is essential for solving the given problem, as we need to differentiate y with respect to t through the intermediate variable x.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, since y is expressed in terms of x, which in turn is expressed in terms of t, implicit differentiation allows us to find dy/dt by treating x as a function of t. This method is particularly useful when dealing with relationships that are not easily solvable for one variable.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. In this problem, y = 3 sin(2x) involves the sine function, which will require knowledge of its properties and derivatives. Understanding how to differentiate trigonometric functions is crucial for finding the derivative of y with respect to t, especially when combined with the Chain Rule.

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