Question

1. if $f(t)=(9t + \frac{2}{t})^{\frac{7}{8}}$, find $f(t)$. $f(t)=$

Explanation:

Step1: Identify the outer - inner functions

Let $u = 9t+\frac{2}{t}=9t + 2t^{-1}$, and $y = u^{\frac{7}{8}}$.

Step2: Differentiate the outer function with respect to $u$

Using the power rule $\frac{d}{du}(u^n)=nu^{n - 1}$, we have $\frac{dy}{du}=\frac{7}{8}u^{\frac{7}{8}-1}=\frac{7}{8}u^{-\frac{1}{8}}$.

Step3: Differentiate the inner function with respect to $t$

$\frac{du}{dt}=\frac{d}{dt}(9t + 2t^{-1})=9-2t^{-2}=9-\frac{2}{t^{2}}$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. Substitute $u = 9t+\frac{2}{t}$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dt}$:
$\frac{dy}{dt}=\frac{7}{8}(9t+\frac{2}{t})^{-\frac{1}{8}}(9 - \frac{2}{t^{2}})$.

Answer:

$\frac{7}{8}(9t+\frac{2}{t})^{-\frac{1}{8}}(9-\frac{2}{t^{2}})$