ind the derivative of the function \( F(x)=-5(3+2 \sqrt{x})^{-6} \)

To find the derivative of the function \( F(x) = -5(3 + 2\sqrt{x})^{-6} \), we can use the chain rule. Start by differentiating the outer function first, which is \( -5u^{-6} \) where \( u = 3 + 2\sqrt{x} \): 1. The derivative of \( -5u^{-6} \) is \( 30u^{-7} \frac{du}{dx} \). 2. Next, we need to find \( \frac{du}{dx} \) where \( u = 3 + 2\sqrt{x} \): - The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \), so \( \frac{du}{dx} = 0 + 2 \cdot \frac{1}{2\sqrt{x}} = \frac{1}{\sqrt{x}} \). Now, combine these results: \[ F'(x) = 30(3 + 2\sqrt{x})^{-7} \cdot \frac{1}{\sqrt{x}} = \frac{30}{\sqrt{x}(3 + 2\sqrt{x})^{7}}. \] So, the derivative is \( F'(x) = \frac{30}{\sqrt{x}(3 + 2\sqrt{x})^{7}} \).