Question

differentiate the function. g(t) = ln(4 + t^6) g(t) =

Explanation:

Step1: Apply chain - rule

Let $u = 4 + t^{6}$, then $g(t)=\ln(u)$. The chain - rule states that $\frac{dg}{dt}=\frac{dg}{du}\cdot\frac{du}{dt}$.
The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$, and $\frac{du}{dt}=\frac{d(4 + t^{6})}{dt}$.

Step2: Differentiate $u$

Differentiating $u = 4 + t^{6}$ with respect to $t$, we get $\frac{du}{dt}=6t^{5}$ (since the derivative of a constant 4 is 0 and the derivative of $t^{n}$ is $nt^{n - 1}$).

Step3: Calculate $g^{\prime}(t)$

Substitute $u = 4 + t^{6}$ and $\frac{du}{dt}=6t^{5}$ into the chain - rule formula. So $g^{\prime}(t)=\frac{1}{4 + t^{6}}\cdot6t^{5}=\frac{6t^{5}}{4 + t^{6}}$.

Answer:

$\frac{6t^{5}}{4 + t^{6}}$