Find the Derivative - d/dx

Problem

d(√(,5*x+√(,5*x+√(,5*x))))/d(x)

Solution

1. Identify the function as a composition of nested functions requiring the chain rule. Let u=5*x+√(,5*x+√(,5*x))

2. Apply the power rule for the outermost square root, which is u(1/2)

d(√(,u))/d(x)=1/(2√(,u))⋅d(u)/d(x)

3. Differentiate the inner expression 5*x+√(,5*x+√(,5*x)) term by term. The derivative of 5*x is 5 For the next nested square root, apply the chain rule again.

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

4. Differentiate the innermost expression 5*x+√(,5*x) The derivative of 5*x is 5 and the derivative of √(,5*x) is 5/(2√(,5*x))

d(5*x+√(,5*x))/d(x)=5+5/(2√(,5*x))

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

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

6. Combine all parts to find the final derivative.

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

Final Answer

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

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