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

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.

### Problem in finding Differentiation of function

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.

### Apply Chain Rule

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.

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

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

### Differentiate the function

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

$= 2x\cos{y}$

### Eliminate the y terms by its replacement

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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