Question

Math

Posted 29 days ago

`What is the second derivative of the function $y = \sin^2 x$?`

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Answer from Sia

Posted 29 days ago

Solution by Steps

step 1

To find the second derivative of the function $y = \sin^2 x$, we first compute the first derivative: $\frac{dy}{dx} = 2 \sin x \cos x = \sin(2x)$

step 2

Next, we differentiate $\sin(2x)$ to find the second derivative: $\frac{d^2y}{dx^2} = 2 \cos(2x)$

step 3

We can also express this as $2 \cos^2 x - 2 \sin^2 x = 2 \cos(2x)$

step 4

Therefore, the second derivative can be simplified to $2 - 4 \sin^2 x$

Answer

The second derivative of $y = \sin^2 x$ is $2 \cos(2x)$ or $2 - 4 \sin^2 x$.

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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