Assume that $x=x(t)$ and $y=y(t)$. Let $y=x^{2}+5$ and $\frac{d x}{d t}=4$ when $x=3$.
Find $\frac{d y}{d t}$ when $x=3$
$\frac{d y}{d t}=24$ (Simplify your answer.)

Step 1 ：We are given that \(y=x^{2}+5\) and \(\frac{d x}{d t}=4\) when \(x=3\). We are asked to find \(\frac{d y}{d t}\) when \(x=3\).

Step 2 ：To find \(\frac{d y}{d t}\), we can use the chain rule of differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 3 ：In this case, the outer function is \(y=x^{2}+5\) and the inner function is \(x\).

Step 4 ：So, we first need to find the derivative of \(y\) with respect to \(x\), denoted as \(\frac{d y}{d x}\), and then multiply it by \(\frac{d x}{d t}\) to get \(\frac{d y}{d t}\).

Step 5 ：The derivative of \(y\) with respect to \(x\) is \(2x\). When \(x=3\), \(\frac{d y}{d x}=2*3=6\).

Step 6 ：Then, we multiply \(\frac{d y}{d x}\) by \(\frac{d x}{d t}\) to get \(\frac{d y}{d t}\).

Step 7 ：So, \(\frac{d y}{d t}=6*4=24\).

Step 8 ：Final Answer: \(\boxed{24}\)