Find the Derivative - d/dx f(x) = 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 the derivative of ln(u) is 1/u⋅d(u)/d(x)

3. Substitute u=3*x into the formula.

4. Differentiate the inner function 3*x with respect to x which results in 3

5. Multiply the outer derivative by the inner derivative.

d(ln(3*x))/d(x)=1/(3*x)⋅3

6. 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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