91Ó°ÊÓ

In this step, we will find the first derivative. The derivative of a function with respect to \(x\) is the rate of change of the function with respect to \(x\). Using the chain rule, we can find the derivative of \(\sin{x^2}\) as follows: $$\frac{dy}{dx}=\frac{d}{dx}(\sin{x^2})$$ $$\frac{dy}{dx}=\cos{x^2}\cdot\frac{d}{dx}(x^2)$$ $$\frac{dy}{dx}=\cos{x^2}\cdot(2x)$$ $$\frac{dy}{dx}=2x\cos{x^2}$$ So, the first derivative, \(\frac{dy}{dx}\), is \(2x\cos{x^2}\).

Now that we have the first derivative, we need to find the second derivative by taking the derivative of the first derivative with respect to \(x\), or \(\frac{d^2y}{dx^2}\). We will need to use the product rule and the chain rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Using the chain rule and the product rule, we have: $$\frac{d^2y}{dx^2}=\frac{d}{dx}(2x\cos{x^2})$$ $$\frac{d^2y}{dx^2}=\frac{d}{dx}(2x)\cos{x^2} + 2x\frac{d}{dx}(\cos{x^2})$$ Now, we find the derivatives: $$\frac{d^2y}{dx^2}=2\cos{x^2} + 2x(-\sin{x^2}\cdot\frac{d}{dx}(x^2))$$ $$\frac{d^2y}{dx^2}=2\cos{x^2} - 4x^2\sin{x^2}$$ So, the second derivative, \(\frac{d^2y}{dx^2}\), is given by \(2\cos{x^2} - 4x^2\sin{x^2}\).