17. - / 1 points differentiate the function. g(t) = ln(7 + t^6) g(t) =

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Let $u = 7 + t^{6}$, so $g(t)=\ln(u)$.

Step2: Differentiate outer function

The derivative of $y=\ln(u)$ with respect to $u$ is $\frac{d}{du}\ln(u)=\frac{1}{u}$.

Step3: Differentiate inner function

The derivative of $u = 7 + t^{6}$ with respect to $t$ is $\frac{du}{dt}=6t^{5}$.

Step4: Apply chain - rule

By the chain - rule, $g^{\prime}(t)=\frac{d}{du}\ln(u)\cdot\frac{du}{dt}$. Substitute $\frac{d}{du}\ln(u)=\frac{1}{u}$ and $\frac{du}{dt}=6t^{5}$ and $u = 7 + t^{6}$ into the formula, we get $g^{\prime}(t)=\frac{6t^{5}}{7 + t^{6}}$.

Answer:

$\frac{6t^{5}}{7 + t^{6}}$