Expert answers to "Use the rules for derivatives to find the..." - Plainmath

f480forever2rz

Answered question

2021-12-03

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

r (t) = \frac{5 t^{2} - 7 t}{{(3 t + 1)}^{3}}

Answer & Explanation

Othrom

Beginner2021-12-04Added 15 answers

Step 1
Consider the given function as

\frac{d}{d t} (\frac{5 t^{2} - 7 t}{{(3 t + 1)}^{3}})

Step 2
Use quotient rule as

{(\frac{f}{g})}^{'} = \frac{f^{'} \cdot g - g^{'} \cdot f}{g^{2}}

Then,

\frac{d}{d t} (\frac{5 t^{2} - 7 t}{{(3 t + 1)}^{3}}) = \frac{\frac{d}{d t} (5 t^{2} - 7 t) {(3 t + 1)}^{3} - \frac{d}{d t} ({(3 t + 1)}^{3}) (5 t^{2} - 7 t)}{{({(3 t + 1)}^{3})}^{2}}

= \frac{(10 t - 7) {(3 t + 1)}^{3} - 9 {(3 t + 1)}^{2} (5 t^{2} - 7 t)}{{({(3 t + 1)}^{3})}^{2}}

= \frac{(10 t - 7) (3 t + 1) - 9 (5 t^{2} - 7 t)}{{({(3 t + 1)}^{3})}^{2}}

= \frac{(30 t^{2} + 10 t - 21 t - 7) - (45 t^{2} - 63 t)}{{({(3 t + 1)}^{3})}^{2}}

= \frac{- 15 t^{2} + 52 t - 7}{{(3 t + 1)}^{4}}

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