**Finding the Derivative of the Function**To find the derivative of the given function, we start with the function provided:\[ f(x) = \sqrt{9 + \sqrt{1 + \sqrt{x}}} \]We need to find the derivative, denoted as $ f'(x) $.1. Let $ u = 1 + \sqrt{x} $.2. Then $ \sqrt{u} = \sqrt{1 + \sqrt{x}} $, and let $ v = \sqrt{u} $. 3. Next, we have $ 9 + v $ inside another root. Let $ w = 9 + v $.The function can now be rewritten in terms of $ w $:\[ f(x) = \sqrt{w} \]The steps to differentiate this function involve using the chain rule multiple times, as well as differentiating square root functions. The completed differentiation process would result in the function $ f'(x) $, which can be placed in the box provided in the image.\[ f'(x) = \]This boxed area serves for the student to either perform their own calculations or check against a step-by-step solution provided in their educational content.No graphs or diagrams are included in the image.

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Description: **Finding the Derivative of the Function**To find the derivative of the given function, we start with the function provided:\[ f(x) = \sqrt{9 + \sqrt{1 + \sqrt{x}}} \]We need to find the derivative, denoted as $ f'(x) $.1. Let \( u = 1 + \sqrt{x}

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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**Finding the Derivative of the Function**

To find the derivative of the given function, we start with the function provided:

\[ f(x) = \sqrt{9 + \sqrt{1 + \sqrt{x}}} \]

We need to find the derivative, denoted as $ f'(x) $.

1. Let $ u = 1 + \sqrt{x} $.
2. Then $ \sqrt{u} = \sqrt{1 + \sqrt{x}} $, and let $ v = \sqrt{u} $.
3. Next, we have $ 9 + v $ inside another root. Let $ w = 9 + v $.

The function can now be rewritten in terms of $ w $:
\[ f(x) = \sqrt{w} \]

The steps to differentiate this function involve using the chain rule multiple times, as well as differentiating square root functions. The completed differentiation process would result in the function $ f'(x) $, which can be placed in the box provided in the image.

\[ f'(x) = \]

This boxed area serves for the student to either perform their own calculations or check against a step-by-step solution provided in their educational content.

No graphs or diagrams are included in the image.

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