Find the Derivative - d/dx natural log of 9x

Problem

d(ln(9*x))/d(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=9*x into the formula, resulting in 1/(9*x)⋅(d(9)*x)/d(x)

4. Differentiate the inner function 9*x with respect to x to get 9

5. Multiply the terms to get 1/(9*x)⋅9

6. Simplify the expression by canceling the common factor of 9 in the numerator and denominator.

Final Answer

d(ln(9*x))/d(x)=1/x

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