Answered: "Use the rules for derivatives to find the..."

Ayaana Buck

Answered question

2021-10-10

Use the rules for derivatives to find the derivative of function defined as follows.

s (t) = \frac{t^{3} - 2 t}{{(4 t - 3)}^{4}}

Answer & Explanation

Nathaniel Kramer

Skilled2021-10-11Added 78 answers

Step 1
Given function is, $s (t) = \frac{t^{3} - 2 t}{{(4 t - 3)}^{4}}$ .
It is differentiated by apply the quotient rule, ${(\frac{f}{g})}^{'} = \frac{f^{'} g - f g^{'}}{g^{2}}$ .
So,
$s^{'} (t) = \frac{d}{d t} (\frac{t^{3} - 2 t}{{(4 t - 3)}^{4}})$
$= \frac{\frac{d}{d t} (t^{3} - 2 t) (4 t - 3)^{4} - \frac{d}{d t} ((4 t - 3)^{4}) (t^{3} - 2 t)}{((4 t - 3)^{4})^{2}}$
Step 2
On further simplifying,
$s^{'} (t) = \frac{\frac{d}{d t} (t^{3} - 2 t) {(4 t - 3)}^{4} - \frac{d}{d t} ({(4 t - 4)}^{4}) (t^{3} - 2 t)}{{({(4 t - 3)}^{4})}^{2}}$
$= \frac{(3 t^{2} - 2) {(4 t - 3)}^{4} - 16 {(4 t - 3)}^{3} (t^{3} - 2 t)}{{({(4 t - 3)}^{4})}^{2}}$
$= \frac{{(4 t - 3)}^{3} ((3 t^{2} - 2) (4 t - 3) - 16 (t^{3} - 2 t))}{{({(4 t - 3)}^{4})}^{2}}$
$= \frac{- 4 t^{3} - 9 t^{2} + 24 t + 6}{{(4 t - 3)}^{5}}$

Jeffrey Jordon

Expert2022-11-15Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

Didn't find what you were looking for?