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(d)/(dx)(sin^(2)(x))

$\frac{d}{d x}\left(\sin ^{2}(x)\right)$

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Evaluate rational exponents

Full solution

Q.

\frac{d}{d x}\left(\sin ^{2}(x)\right)

Recognize Function: Step $1$ : Recognize the function to differentiate. $\newline$ We need to find the derivative of $\sin^2(x)$ .
Apply Chain Rule: Step $2$ : Apply the chain rule for differentiation. $\newline$ Let $u = \sin(x)$ , then we need to find the derivative of $u^2$ . $\newline$ $\frac{d}{dx}(u^2) = 2u \cdot \frac{du}{dx}$ .
Substitute and Simplify: Step $3$ : Substitute back for $u$ and $\frac{du}{dx}$ .
$u = \sin(x)$ and $\frac{du}{dx} = \cos(x)$ .
So, $\frac{d}{dx}(\sin^2(x)) = 2\sin(x) \cdot \cos(x)$ .
Substitute and Simplify: Step $3$ : Substitute back for $u$ and $\frac{du}{dx}$ . $u = \sin(x)$ and $\frac{du}{dx} = \cos(x)$ .So, $\frac{d}{dx}(\sin^2(x)) = 2\sin(x) \cdot \cos(x)$ . Step $4$ : Simplify the expression. $2\sin(x)\cos(x)$ can be written as $\sin(2x)$ using the double angle formula for sine.

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