Question
Find \( \frac{d s}{d t} \) if \( s=\frac{t}{7 t+3} \) \( \frac{d s}{d t}=\square \)
Ask by Henry Mckenzie. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{ds}{dt} = \frac{3}{(7t+3)^2}
\]
Solution
We are given
\[
s = \frac{t}{7t+3}.
\]
To find \(\frac{ds}{dt}\), we use the quotient rule. The quotient rule states that if
\[
s = \frac{u(t)}{v(t)},
\]
then
\[
\frac{ds}{dt} = \frac{v(t) \cdot \frac{d}{dt}[u(t)] - u(t) \cdot \frac{d}{dt}[v(t)]}{[v(t)]^2}.
\]
Let
\[
u(t) = t \quad \text{and} \quad v(t) = 7t+3.
\]
First, compute the derivatives:
\[
\frac{d}{dt}[u(t)] = 1 \quad \text{and} \quad \frac{d}{dt}[v(t)] = 7.
\]
Now substitute into the quotient rule formula:
\[
\frac{ds}{dt} = \frac{(7t+3)(1) - t(7)}{(7t+3)^2}.
\]
Simplify the numerator:
\[
(7t+3) - 7t = 3.
\]
Thus, we have:
\[
\frac{ds}{dt} = \frac{3}{(7t+3)^2}.
\]
So, the derivative is:
\[
\frac{ds}{dt} = \boxed{\frac{3}{(7t+3)^2}}.
\]
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Simplify this solution Beyond the Answer
To find \( \frac{d s}{d t} \) for the function \( s = \frac{t}{7t + 3} \), we can use the quotient rule of differentiation. The quotient rule states that if you have a function \( s = \frac{u}{v} \), then the derivative is given by: \[ \frac{d s}{d t} = \frac{u'v - uv'}{v^2} \] Here, \( u = t \) and \( v = 7t + 3 \). Calculating \( u' \) and \( v' \): - \( u' = 1 \) - \( v' = 7 \) Now, we can apply the quotient rule: \[ \frac{d s}{d t} = \frac{(1)(7t + 3) - (t)(7)}{(7t + 3)^2} \] Simplifying the numerator: \[ 7t + 3 - 7t = 3 \] Thus: \[ \frac{d s}{d t} = \frac{3}{(7t + 3)^2} \] So, the final result is: \[ \frac{d s}{d t} = \frac{3}{(7t + 3)^2} \]
