Question

1. y = sec(8πt)

Explanation:

Step1: Recall the derivative formula for $\sec(u)$

The derivative of $\sec(u)$ with respect to $u$ is $\sec(u)\tan(u)$. Also, by the chain - rule, if $y = f(g(t))$, then $y^\prime=f^\prime(g(t))\cdot g^\prime(t)$. Here $u = 8\pi t$ and $y=\sec(u)$.

Step2: Find the derivative of the inner function

The derivative of $u = 8\pi t$ with respect to $t$ is $u^\prime=\frac{d}{dt}(8\pi t)=8\pi$.

Step3: Apply the chain - rule

The derivative of $y=\sec(8\pi t)$ with respect to $t$ is $y^\prime=\sec(8\pi t)\tan(8\pi t)\cdot8\pi$.

Answer:

$y^\prime = 8\pi\sec(8\pi t)\tan(8\pi t)$