Find the Derivative - d/dx square root of natural log of 3x

Problem

d(√(,ln(3*x)))/d(x)

Solution

1. Identify the outer function and the inner function to prepare for the chain rule. The expression is of the form u(1/2) where u=ln(3*x)

2. Apply the power rule to the square root function. The derivative of √(,u) is 1/(2√(,u))

3. Apply the chain rule by multiplying by the derivative of the inner function ln(3*x)

d(√(,ln(3*x)))/d(x)=1/(2√(,ln(3*x)))⋅d(ln(3*x))/d(x)

4. Differentiate the natural log function using the chain rule again. The derivative of ln(3*x) is 1/(3*x)⋅3

d(ln(3*x))/d(x)=1/(3*x)⋅3

5. Simplify the derivative of the inner function.

1/(3*x)⋅3=1/x

6. Combine all parts to find the final derivative.

1/(2√(,ln(3*x)))⋅1/x=1/(2*x√(,ln(3*x)))

Final Answer

d(√(,ln(3*x)))/d(x)=1/(2*x√(,ln(3*x)))

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