**Problem Statement:**Use the quotient rule to find the derivative of the function.\[ p(t) = \frac{\sqrt{t}}{2t - 3} \]**Solution:**Find the derivative $ p'(t) $ using the quotient rule. The quotient rule states that for a function $\frac{u}{v}$, the derivative is:\[ \frac{d}{dt}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]**Function Details:**- **Numerator (u):** $\sqrt{t}$- **Denominator (v):** $2t - 3$**Steps:**1. Compute the derivative of the numerator $u(t)$: $ u(t) = \sqrt{t} = t^{1/2} $ Derivative $ u'(t) = \frac{1}{2}t^{-1/2} = \frac{1}{2\sqrt{t}} $2. Compute the derivative of the denominator $v(t)$: $ v(t) = 2t - 3 $ Derivative $ v'(t) = 2 $3. Apply the quotient rule formula: \[ p'(t) = \frac{(2t - 3) \cdot \frac{1}{2\sqrt{t}} - \sqrt{t} \cdot 2}{(2t - 3)^2} \]Fill in the derivative in the provided box in mathematical form.

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Description: **Problem Statement:**Use the quotient rule to find the derivative of the function.\[ p(t) = \frac{\sqrt{t}}{2t - 3} \]**Solution:**Find the derivative $ p'(t) $ using the quotient rule. The quotient rule states that for a function $\frac{u}{v}$

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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**Problem Statement:**

Use the quotient rule to find the derivative of the function.

\[ p(t) = \frac{\sqrt{t}}{2t - 3} \]

**Solution:**

Find the derivative $ p'(t) $ using the quotient rule. The quotient rule states that for a function $\frac{u}{v}$, the derivative is:

\[ \frac{d}{dt}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]

**Function Details:**

- **Numerator (u):** $\sqrt{t}$
- **Denominator (v):** $2t - 3$

**Steps:**

1. Compute the derivative of the numerator $u(t)$:
$ u(t) = \sqrt{t} = t^{1/2} $
Derivative $ u'(t) = \frac{1}{2}t^{-1/2} = \frac{1}{2\sqrt{t}} $

2. Compute the derivative of the denominator $v(t)$:
$ v(t) = 2t - 3 $
Derivative $ v'(t) = 2 $

3. Apply the quotient rule formula:
\[ p'(t) = \frac{(2t - 3) \cdot \frac{1}{2\sqrt{t}} - \sqrt{t} \cdot 2}{(2t - 3)^2} \]

Fill in the derivative in the provided box in mathematical form.

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