uotient Rule to determine the derivative of the function \( G(x)=\frac{2 x+5}{x-8} \)

Evaluate the derivative by following steps: - step0: Solution: \(\frac{d}{dx}\left(\frac{2x+5}{x-8}\right)\) - step1: Use differentiation rules: \(\frac{\frac{d}{dx}\left(2x+5\right)\times \left(x-8\right)-\left(2x+5\right)\times \frac{d}{dx}\left(x-8\right)}{\left(x-8\right)^{2}}\) - step2: Calculate: \(\frac{2\left(x-8\right)-\left(2x+5\right)\times 1}{\left(x-8\right)^{2}}\) - step3: Multiply: \(\frac{2\left(x-8\right)-\left(2x+5\right)}{\left(x-8\right)^{2}}\) - step4: Subtract the terms: \(\frac{-21}{\left(x-8\right)^{2}}\) - step5: Rewrite the fraction: \(-\frac{21}{\left(x-8\right)^{2}}\) - step6: Calculate: \(-\frac{21}{x^{2}-16x+64}\) To find the derivative of the function \( G(x) = \frac{2x + 5}{x - 8} \) using the Quotient Rule, we follow these steps: 1. **Identify the functions**: - Let \( u = 2x + 5 \) (the numerator) - Let \( v = x - 8 \) (the denominator) 2. **Apply the Quotient Rule**: The Quotient Rule states that if you have a function \( G(x) = \frac{u}{v} \), then the derivative \( G'(x) \) is given by: \[ G'(x) = \frac{u'v - uv'}{v^2} \] where \( u' \) is the derivative of \( u \) and \( v' \) is the derivative of \( v \). 3. **Calculate the derivatives**: - \( u' = \frac{d}{dx}(2x + 5) = 2 \) - \( v' = \frac{d}{dx}(x - 8) = 1 \) 4. **Substitute into the Quotient Rule**: \[ G'(x) = \frac{(2)(x - 8) - (2x + 5)(1)}{(x - 8)^2} \] 5. **Simplify the expression**: - The numerator becomes: \[ 2(x - 8) - (2x + 5) = 2x - 16 - 2x - 5 = -21 \] - Therefore, we have: \[ G'(x) = \frac{-21}{(x - 8)^2} \] Thus, the derivative of the function \( G(x) \) is: \[ G'(x) = -\frac{21}{(x - 8)^2} \]