LeSSOn20 3.7 The chain rule: Probien (1 point) Find the derivative of \( y \) using the appropriate combination of rules. \( y=(9 x+4)^{\frac{1}{2}} \) \( y^{\prime}=\square \) Preview My Answers Submit Answers

1. Identify the outer function and the inner function: - Outer function: \( u^{\frac{1}{2}} \) where \( u = 9x+4 \) - Inner function: \( u = 9x + 4 \) 2. Differentiate the outer function with respect to \( u \): \[ \frac{d}{du}\left(u^{\frac{1}{2}}\right) = \frac{1}{2} u^{-\frac{1}{2}} \] 3. Differentiate the inner function \( u = 9x + 4 \) with respect to \( x \): \[ \frac{du}{dx} = 9 \] 4. Apply the chain rule by multiplying the derivatives: \[ y' = \frac{1}{2} u^{-\frac{1}{2}} \cdot 9 = \frac{9}{2} (9x+4)^{-\frac{1}{2}} \] 5. The derivative is: \[ y' = \frac{9}{2} (9x+4)^{-\frac{1}{2}} \]