Find the Derivative - d/dx y = natural log of 3x

Problem

d()/d(x)*ln(3*x)

Solution

1. Identify the function as a composition of the natural logarithm and a linear function, which requires the use of the chain rule.

2. Apply the chain rule for the natural logarithm, which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Substitute u=3*x and find its derivative, which is (d(3)*x)/d(x)=3

4. Combine the results to get the derivative of the outer function times the derivative of the inner function.

d(ln(3*x))/d(x)=1/(3*x)⋅3

5. Simplify the expression by canceling the constant factor of 3 in the numerator and denominator.

3/(3*x)=1/x

Final Answer

d(ln(3*x))/d(x)=1/x

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