use logarithmic differentiation to find the derivative of y with respect to t. y = (\\sqrt7{t})^t…

Explanation:

Step1: Rewrite the function

First, rewrite $y = (\sqrt[7]{t})^t=t^{\frac{t}{7}}$. Then take the natural - logarithm of both sides: $\ln y=\ln(t^{\frac{t}{7}})=\frac{t}{7}\ln t$.

Step2: Differentiate both sides with respect to $t$

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = \frac{t}{7}$ and $v=\ln t$. The derivative of $u$ with respect to $t$ is $\frac{1}{7}$, and the derivative of $v$ with respect to $t$ is $\frac{1}{t}$. So, $\frac{1}{y}\frac{dy}{dt}=\frac{1}{7}\ln t+\frac{t}{7}\cdot\frac{1}{t}=\frac{1}{7}\ln t+\frac{1}{7}$.

Step3: Solve for $\frac{dy}{dt}$

Multiply both sides by $y = t^{\frac{t}{7}}$ to get $\frac{dy}{dt}=t^{\frac{t}{7}}\left(\frac{1}{7}\ln t+\frac{1}{7}\right)=\frac{1}{7}t^{\frac{t}{7}}(1 + \ln t)$.

Answer:

$\frac{1}{7}t^{\frac{t}{7}}(1+\ln t)$