Find the derivative of $ sin(x^2) $ with respect to $ x $

To find the derivative of $ \sin(x^2) $ with respect to $ x $, we use the chain rule:

$$ \frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx} $$

Here, let $ u = x^2 $. Then:

$$ \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x $$

Now apply the chain rule:

$$ \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x $$

The final derivative is:

$$ \frac{d}{dx} \sin(x^2) = 2x \cos(x^2) $$

To find the derivative of $ \sin(x^2) $, use the chain rule:

$$ \frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx} $$

Let $ u = x^2 $, so:

$$ \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x $$

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