Find the derivative of the function given by $f(x)=\sin \left(x^{2}\right)$.

Step 1: Identify the function and rule

The given function is $f(x) = \sin(x^2)$. This is a composite function, meaning it's a function within a function. To differentiate such functions, we need to use the chain rule.

Step 2: Apply the Chain Rule

The chain rule states that the derivative of a composite function $f(g(x))$ is the derivative of the outer function $f$ evaluated at the inner function $g(x)$, multiplied by the derivative of the inner function $g(x)$. In our case, the outer function is $\sin(u)$ and the inner function is $u = x^2$.

Step 3: Differentiate the outer function

First, we differentiate the outer function, $\sin(u)$, with respect to $u$. The derivative of $\sin(u)$ is $\cos(u)$.

Step 4: Differentiate the inner function

Next, we differentiate the inner function, $x^2$, with respect to $x$. Using the power rule, the derivative of $x^2$ is $2x$.

Step 5: Combine the derivatives

Finally, we combine the results from differentiating the outer and inner functions according to the chain rule. We substitute $u = x^2$ back into the derivative of the outer function and multiply it by the derivative of the inner function.

Step 6: Simplify the expression

Rearranging the terms gives us the final simplified derivative of the function.

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