Find the Derivative - d/dx

Problem

d()/d(x)√(,11*x+√(,11*x+√(,11*x)))

Solution

1. Identify the function as a composition of nested functions requiring the chain rule, where the outermost function is u(1/2) with u=11*x+√(,11*x+√(,11*x))

2. Apply the power rule to the outermost square root to find the first part of the derivative.

d(y)/d(x)=1/(2√(,11*x+√(,11*x+√(,11*x))))⋅d()/d(x)*(11*x+√(,11*x+√(,11*x)))

3. Differentiate the inner expression using the sum rule and the chain rule again for the second square root.

d()/d(x)*(11*x+√(,11*x+√(,11*x)))=11+1/(2√(,11*x+√(,11*x)))⋅d()/d(x)*(11*x+√(,11*x))

4. Differentiate the innermost expression.

d()/d(x)*(11*x+√(,11*x))=11+11/(2√(,11*x))

5. Substitute the inner derivatives back into the chain rule structure.

d(y)/d(x)=1/(2√(,11*x+√(,11*x+√(,11*x))))⋅(11+(11+11/(2√(,11*x)))/(2√(,11*x+√(,11*x))))

6. Simplify the expression by distributing the terms.

d(y)/d(x)=11/(2√(,11*x+√(,11*x+√(,11*x))))+(11+11/(2√(,11*x)))/(4√(,11*x+√(,11*x+√(,11*x)))√(,11*x+√(,11*x)))

Final Answer

d(√(,11*x+√(,11*x+√(,11*x))))/d(x)=11/(2√(,11*x+√(,11*x+√(,11*x))))*(1+1/(2√(,11*x+√(,11*x)))*(1+1/(2√(,11*x))))

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