If y = sin(2x), find dy/dx at x = π/4.

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Question: If y = sin(2x), find dy/dx at x = π/4.

Options:

Correct Answer: √2

Solution:

dy/dx = 2cos(2x). At x = π/4, dy/dx = 2cos(π/2) = 2(0) = 0.

If y = sin(2x), find dy/dx at x = π/4.

If y = sin(2x), find dy/dx at x = π/4.

Correct Answer: 0

Step 1: Identify the function given in the question. Here, y = sin(2x).
Step 2: To find dy/dx, we need to differentiate y with respect to x. The derivative of sin(u) is cos(u) * du/dx, where u = 2x.
Step 3: Differentiate y = sin(2x). The derivative dy/dx = cos(2x) * d(2x)/dx.
Step 4: Since d(2x)/dx = 2, we have dy/dx = 2 * cos(2x).
Step 5: Now, we need to find dy/dx at x = π/4. Substitute x = π/4 into the derivative: dy/dx = 2 * cos(2 * (π/4)).
Step 6: Calculate 2 * (π/4) = π/2. So, we need to find cos(π/2).
Step 7: The value of cos(π/2) is 0.
Step 8: Now substitute this value back into the derivative: dy/dx = 2 * 0 = 0.

Differentiation of Trigonometric Functions – The question tests the ability to differentiate the sine function, particularly when it involves a composite function with a coefficient.
Chain Rule – The use of the chain rule is essential when differentiating functions like sin(2x), where the inner function (2x) must be considered.