(1 point) Suppose that $x=x(t)$ and $y=y(t)$ are both functions of $t$. If
\[
y=4 x^{2}-1,
\]
and $d x / d t=4$ when $x=19$, what is $d y / d t$ ?
\[
d y / d t=
\]

Answer

Expert–verified

Hide Steps

Answer

$\boxed{\frac{dy}{dt} = 608}$ when $x = 19$.

Steps

Step 1 ：Given the function $y = 4x^2 - 1$, we can differentiate it with respect to $x$ to get $\frac{dy}{dx} = 8x$.

Step 2 ：We know that $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$.

Step 3 ：We are given that $\frac{dx}{dt} = 4$ when $x = 19$.

Step 4 ：Substitute these values into the equation to find $\frac{dy}{dt}$: $\frac{dy}{dt} = 8x \cdot \frac{dx}{dt} = 8\cdot19\cdot4 = 608$.

Step 5 ：$\boxed{\frac{dy}{dt} = 608}$ when $x = 19$.