Find the Derivative - d/dx y = square root of natural log of 2x

Problem

d()/d(x)√(,ln(2*x))

Solution

1. Identify the outer function as a square root, which can be rewritten as an exponent of 1/2

d()/d(x)*(ln(2*x))1/2

2. Apply the power rule and the chain rule to the outer function.

1/2*(ln(2*x))(−1/2)⋅d(ln(2*x))/d(x)

3. Apply the chain rule to the natural log function, where the inner function is 2*x

1/2*(ln(2*x))(−1/2)⋅1/(2*x)⋅(d(2)*x)/d(x)

4. Differentiate the innermost function 2*x with respect to x

1/2*(ln(2*x))(−1/2)⋅1/(2*x)⋅2

5. Simplify the expression by canceling the constants and rewriting the negative exponent as a square root in the denominator.

1/(2*x√(,ln(2*x)))

Final Answer

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

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