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

Problem

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

Solution

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

2. Apply the chain rule which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x) where u=3*x

3. Differentiate the inner function u=3*x with respect to x which gives (d(3)*x)/d(x)=3

4. Substitute these values into the chain rule formula.

d(ln(3*x))/d(x)=1/(3*x)⋅3

5. Simplify the expression by canceling the common 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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