Actually, $\sin{x}$ is a trigonometric function and $x^2$ is an exponential function in algebraic form. They both formed a special function by the composition of them.

According to the derivative of $\sin{x}$ with respect to $x$ formula, the derivative of $\sin{x}$ with respect to $x$ is equal to $\cos{x}$.

$\dfrac{d}{dx}{\sin{x}} = \cos{x}$

This formula cannot be applied directly to this derivative problem due to the angle difference of the trigonometric function.

Chain rule is only one solution to deal functions which are formed by the composition of two or more functions.

$\dfrac{d}{dx}{\sin{(x^2)}}$

Take $y = x^2$, then $\dfrac{dy}{dx} = 2x$, therefore $dy = 2xdx$ and then $dx = \dfrac{dy}{2x}$. Now, transform the whole differential function by this data.

$= \dfrac{d}{\dfrac{dy}{2x}}{\sin{y}}$

$= 2x\dfrac{d}{dy}{\sin{y}}$

Now, differentiate the sine function with respect to $y$.

$= 2x\cos{y}$

Actually, $y = x^2$. So, replace the term $y$ by its replacement for obtaining the required result of this differentiation problem in calculus.

$\therefore \,\,\,\,\,\, \dfrac{d}{dx}{\sin{(x^2)}} = 2x\cos{x^2}$

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