Question

use the quotient rule to find the derivative of the following.

\\( y = \frac{6t - 5}{2t + 1} \\)

the solution is \\( \frac{dy}{dt} = \square \\).

Explanation:

Step1: Recall Quotient Rule

The quotient rule states that if \( y = \frac{u}{v} \), then \( \frac{dy}{dt} = \frac{u'v - uv'}{v^2} \), where \( u = 6t - 5 \) and \( v = 2t + 1 \).

Step2: Find \( u' \) and \( v' \)

• For \( u = 6t - 5 \), the derivative \( u' = \frac{d}{dt}(6t - 5) = 6 \).
• For \( v = 2t + 1 \), the derivative \( v' = \frac{d}{dt}(2t + 1) = 2 \).

Step3: Substitute into Quotient Rule

Substitute \( u = 6t - 5 \), \( u' = 6 \), \( v = 2t + 1 \), and \( v' = 2 \) into the quotient rule formula:

$$ \frac{dy}{dt} = \frac{(6)(2t + 1) - (6t - 5)(2)}{(2t + 1)^2} $$

Step4: Simplify the Numerator

• Expand the numerator: \( 6(2t + 1) - 2(6t - 5) = 12t + 6 - 12t + 10 \).
• Combine like terms: \( 12t - 12t + 6 + 10 = 16 \).

Step5: Write the Final Derivative

The simplified derivative is \( \frac{dy}{dt} = \frac{16}{(2t + 1)^2} \).

Answer:

\(\frac{16}{(2t + 1)^2}\)