Find the Derivative - d/dx

Problem

d()/d(x)√(,5*x+√(,5*x+√(,5*x)))

Solution

1. Identify the function as a composition of nested square roots, which requires the repeated application of the chain rule.

2. Apply the chain rule to the outermost square root, using the rule d(√(,u))/d(x)=1/(2√(,u))⋅d(u)/d(x)

d(√(,5*x+√(,5*x+√(,5*x))))/d(x)=1/(2√(,5*x+√(,5*x+√(,5*x))))⋅d(5*x+√(,5*x+√(,5*x)))/d(x)

3. Differentiate the inner expression 5*x+√(,5*x+√(,5*x)) term by term.

d(5*x+√(,5*x+√(,5*x)))/d(x)=5+d(√(,5*x+√(,5*x)))/d(x)

4. Apply the chain rule again to the second nested square root.

d(√(,5*x+√(,5*x)))/d(x)=1/(2√(,5*x+√(,5*x)))⋅d(5*x+√(,5*x))/d(x)

5. Differentiate the innermost expression 5*x+√(,5*x)

d(5*x+√(,5*x))/d(x)=5+5/(2√(,5*x))

6. Substitute the results back into the chain rule structure.

d(y)/d(x)=1/(2√(,5*x+√(,5*x+√(,5*x))))⋅(5+1/(2√(,5*x+√(,5*x)))⋅(5+5/(2√(,5*x))))

7. Simplify the expression by distributing the terms.

d(y)/d(x)=5/(2√(,5*x+√(,5*x+√(,5*x))))+(5+5/(2√(,5*x)))/(4√(,5*x+√(,5*x+√(,5*x)))√(,5*x+√(,5*x)))

Final Answer

d(√(,5*x+√(,5*x+√(,5*x))))/d(x)=5/(2√(,5*x+√(,5*x+√(,5*x))))+5/(4√(,5*x+√(,5*x+√(,5*x)))√(,5*x+√(,5*x)))+5/(8√(,5*x+√(,5*x+√(,5*x)))√(,5*x+√(,5*x))√(,5*x))

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