**Title: Differentiating the Function \( f(x) = \sqrt{x} + \sqrt{\sqrt{x}} \)**---**Objective:**Differentiate the function \( f(x) = \sqrt{x} + \sqrt{\sqrt{x}} \).**Steps:**1. **Rewrite the Function:** - Given \( f(x) = \sqrt{x} + \sqrt{\sqrt{x}} \). - Rewrite as \( f(x) = (x + x^{1/2})^{1/2} \) or \( (x + (x^{1/2}))^{1/2} \).2. **Apply the Chain Rule:** - Start with the derivative of the outer function: \[ \frac{d}{dx} (x + x^{1/2})^{1/2} = \frac{1}{2} (x + (x^{1/2}))^{-1/2} \cdot \frac{d}{dx} (x + (x^{1/2})) \]3. **Derivatives Inside the Chain Rule:** - Use the chain rule: \[ \frac{1}{2} \left(\frac{1}{\sqrt{x + \sqrt{x}}} \right) \cdot \left( \frac{d}{dx} (x) + \frac{d}{dx} (x^{1/2}) \right) \] - Differentiate each component: - \( \frac{d}{dx}(x) = 1 \) - \( \frac{d}{dx}(x^{1/2}) = \frac{1}{2\sqrt{x}} \)4. **Combine and Simplify:** - Now combine: \[ \frac{1}{2\sqrt{x+\sqrt{x}}} \cdot \left( 1 + \frac{1}{2\sqrt{x}} \right) \]5. **Final Expression:** - The simplified expression becomes: \[ \frac{1}{2\sqrt{x+\sqrt{x}}} \cdot \left( \frac{1}{2\sqrt{x}} \right) \] - Which simplifies to: \[ \frac{1}{2\sqrt{x+\sqrt{x}}} \cd

Name: **Title: Differentiating the Function \( f(x) = \sqrt{x} + \sqrt{\sqr
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Title: Differentiating the Function \( f(x) = \sqrt{x} + \sqrt{\sqrt{x}} \)**---**Objective:**Differentiate the function \( f(x) = \sqrt{x} + \sqrt{\sqrt{x}} \).**Steps:**1. **Rewrite the Function:** - Given \( f(x) = \sqrt{x} + \sqrt{\sqrt{x}}