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Question
Math
Posted 6 months ago
What is the second derivative of the function y=sin2x y = \sin^2 x ?
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Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
To find the second derivative of the function y=sin2x y = \sin^2 x , we first compute the first derivative: dydx=2sinxcosx=sin(2x) \frac{dy}{dx} = 2 \sin x \cos x = \sin(2x)
step 2
Next, we differentiate sin(2x) \sin(2x) to find the second derivative: d2ydx2=2cos(2x) \frac{d^2y}{dx^2} = 2 \cos(2x)
step 3
We can also express this as 2cos2x2sin2x=2cos(2x) 2 \cos^2 x - 2 \sin^2 x = 2 \cos(2x)
step 4
Therefore, the second derivative can be simplified to 24sin2x 2 - 4 \sin^2 x
Answer
The second derivative of y=sin2x y = \sin^2 x is 2cos(2x) 2 \cos(2x) or 24sin2x 2 - 4 \sin^2 x .
Key Concept
The second derivative measures the curvature of the function, indicating how the slope of the tangent line changes.
Explanation
The second derivative provides information about the concavity of the function, which is essential for understanding its behavior.

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