Use the product or quotient rule or the generalized power rule to find the derivative of the function. \( y y=(6 x+5)^{6} \sqrt{x} \)

To find the derivative of the function \( y = (6x + 5)^{6} \sqrt{x} \), we will use the product rule, which states that if you have a function \( y = u \cdot v \), then the derivative \( y' \) is given by: \[ y' = u'v + uv' \] In this case, we can identify: - \( u = (6x + 5)^{6} \) - \( v = \sqrt{x} = x^{1/2} \) Next, we need to find the derivatives \( u' \) and \( v' \). 1. **Finding \( u' \)** using the chain rule: \[ u' = 6(6x + 5)^{5} \cdot (6) = 36(6x + 5)^{5} \] 2. **Finding \( v' \)**: \[ v' = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \] Now we can apply the product rule: \[ y' = u'v + uv' \] Substituting \( u \), \( u' \), \( v \), and \( v' \) into the product rule formula: \[ y' = 36(6x + 5)^{5} \cdot \sqrt{x} + (6x + 5)^{6} \cdot \frac{1}{2\sqrt{x}} \] Now, let's simplify this expression. The first term is: \[ 36(6x + 5)^{5} \cdot \sqrt{x} \] The second term is: \[ \frac{(6x + 5)^{6}}{2\sqrt{x}} \] To combine these terms, we can express them with a common denominator. The common denominator will be \( 2\sqrt{x} \): \[ y' = \frac{72(6x + 5)^{5} \cdot x + (6x + 5)^{6}}{2\sqrt{x}} \] Now, we can factor out \( (6x + 5)^{5} \): \[ y' = \frac{(6x + 5)^{5} (72x + (6x + 5))}{2\sqrt{x}} \] Simplifying the expression inside the parentheses: \[ 72x + 6x + 5 = 78x + 5 \] Thus, the derivative is: \[ y' = \frac{(6x + 5)^{5} (78x + 5)}{2\sqrt{x}} \] This is the final result for the derivative of the function \( y = (6x + 5)^{6} \sqrt{x} \).