Given $y=4 x^{2}+x$, find $\frac{d y}{d t}$ when $x=-5$ and $\frac{d x}{d t}=3$ $\frac{d y}{d t}=\square$ (Simplify your answer.)

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

Step 2 ：To find \(\frac{d y}{d t}\), we need to use the chain rule of differentiation which 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 \(y\) and the inner function is \(x\). So, we first need to find \(\frac{d y}{d x}\) and then multiply it by \(\frac{d x}{d t}\) to get \(\frac{d y}{d t}\).

Step 4 ：First, we find the derivative of \(y\) with respect to \(x\), denoted as \(\frac{d y}{d x}\). For the function \(y=4 x^{2}+x\), the derivative \(\frac{d y}{d x}\) is \(8x + 1\).

Step 5 ：We are given that \(\frac{d x}{d t}=3\).

Step 6 ：Then, we find \(\frac{d y}{d t}\) by multiplying \(\frac{d y}{d x}\) by \(\frac{d x}{d t}\). This gives us \(\frac{d y}{d t} = (8x + 1) * 3 = 24x + 3\).

Step 7 ：We are asked to find \(\frac{d y}{d t}\) when \(x=-5\). Substituting \(x=-5\) into the equation for \(\frac{d y}{d t}\), we get \(\frac{d y}{d t} = 24*(-5) + 3 = -117\).

Step 8 ：So, the final answer is \(\frac{d y}{d t}=\boxed{-117}\) when \(x=-5\) and \(\frac{d x}{d t}=3\).