Differentiate the following functions of \(y\) with respect to \(x\) : (a) \((y+3)^{4}\) (b) \(\left(y^{2}+3\right)^{4}\)

First, we need to identify the inner and outer functions for each of the given functions, as we will need them for applying the chain rule. (a) \((y+3)^{4}\): here, the inner function is \(y+3\) and the outer function is \(u^4\) where \(u=y+3\). (b) \(\left(y^{2}+3\right)^{4}\): here, the inner function is \(y^2+3\) and the outer function is \(u^4\) where \(u=y^2+3\).

Now, we will apply the chain rule to find the derivatives of the given functions. Remember, the chain rule states that if we have a composite function \(h(x) = f(g(x))\), then the derivative of \(h\) with respect to \(x\) is \(h'(x) = f'(g(x))g'(x)\). (a) \((y+3)^{4}\): Using the chain rule, we have \(\frac{d}{dx}((y+3)^4)=\frac{d}{dy}(u^4)\cdot \frac{dy}{dx}\) Since, \(u = y+3\), we get \(\frac{d}{dy}(u^4) = 4u^3\), so the derivative is \(4(y+3)^3\cdot \frac{dy}{dx}\). (b) \(\left(y^{2}+3\right)^{4}\): Using the chain rule, we have \(\frac{d}{dx}((y^{2}+3)^4)=\frac{d}{dy}(u^4)\cdot \frac{dy}{dx}\) Since, \(u = y^2+3\), we get \(\frac{d}{dy}(u^4) = 4u^3\), so the derivative is \(4(y^2+3)^3\cdot \frac{dy}{dx}\).

Now, we can express the final answer for the derivatives: (a) \(\frac{d}{dx}((y+3)^4) = 4(y+3)^3\cdot \frac{dy}{dx}\) (b) \(\frac{d}{dx}((y^{2}+3)^4) = 4(y^2+3)^3\cdot \frac{dy}{dx}\) Keep in mind that the derivatives are represented in terms of \(\frac{dy}{dx}\), as we are differentiating the functions of \(y\) with respect to \(x\).