se the Quotient Rule to determine the derivative of the function \( w(s)=\frac{2+5 e^{s}}{6 e^{s}+5} \)

To find the derivative of \( w(s) = \frac{2 + 5e^s}{6e^s + 5} \) using the Quotient Rule, we define the numerator \( u(s) = 2 + 5e^s \) and the denominator \( v(s) = 6e^s + 5 \). The Quotient Rule states that \( w'(s) = \frac{u'v - uv'}{v^2} \). First, calculate \( u' \) and \( v' \): - \( u' = 5e^s \) (the derivative of \( 2 + 5e^s \)) - \( v' = 6e^s \) (the derivative of \( 6e^s + 5 \)) Now substitute into the Quotient Rule formula: \[ w'(s) = \frac{(5e^s)(6e^s + 5) - (2 + 5e^s)(6e^s)}{(6e^s + 5)^2} \] Next, simplify the numerator: 1. Expand: \( 5e^s(6e^s + 5) = 30e^{2s} + 25e^s \) 2. Expand: \( (2 + 5e^s)(6e^s) = 12e^s + 30e^{2s} \) 3. Combine terms: \[ 30e^{2s} + 25e^s - (12e^s + 30e^{2s}) = 13e^s \] Therefore, the derivative becomes: \[ w'(s) = \frac{13e^s}{(6e^s + 5)^2} \] This gives you the desired derivative!