Find the derivative of the function:

\[ y = \sqrt{3x + \sqrt{3x + \sqrt{3x}}} \]

y = 3 x + 3 x + 3 x ​ ​ ​ y ′ = ( 3 x + 3 x + 3 x ​ ​ ​ ) ′ = 2 3 x + 3 x + 3 x ​ ​ ​ 1 ​ ⋅ [ 3 + 2 3 x + 3 x ​ ​ 1 ​ ⋅ ( 3 + 2 3 x ​ 1 ​ ⋅ 3 ) ] = 2 3 x + 3 x + 3 x ​ ​ ​ 1 ​ ⋅ [ 3 + 2 3 x + 3 x ​ ​ 1 ​ ⋅ 2 3 x ​ 6 3 x ​ + 3 ​ ]
= 2 3 x + 3 x + 3 x ​ ​ ​ 1 ​ ⋅ [ 3 + 4 9 x 2 + 3 x 3 x ​ ​ 6 3 x ​ + 3 ​ ] = 2 3 x + 3 x + 3 x ​ ​ ​ 1 ​ ⋅ 4 9 x 2 + 3 x 3 x ​ ​ 12 9 x 2 + 3 x 3 x ​ ​ + 6 3 x ​ + 3 ​ = 8 27 x 3 + 9 x 2 3 x ​ + 9 x 2 3 x + 3 x ​ ​ + 3 x 9 x 2 + 3 x 3 x ​ ​ ​ 12 9 x 2 + 3 x 3 x ​ ​ + 6 3 x ​ + 3 ​

Answered by Anonymous | 2024-06-10

The **derivation **of the function y = 3 x + 3 x + 3 x ​ ​ ​ involves the chain rule and the general power rule of differentiation. Due to the complexity of the function, this derivative will be quite complicated, requiring triple application of the chain rule. It's important to verify if the expression inside the square roots are correct. ;

Answered by MonteBlue | 2024-06-18

The derivative of the function y = 3 x + 3 x + 3 x ​ ​ ​ can be found using the chain rule. First, identify the outer and inner functions and apply the derivatives accordingly. Finally, simplify to find the derivative in a manageable form.
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Answered by Anonymous | 2024-09-30