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

Problem

d()/d(x)*ln(9*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. Set u=9*x and find its derivative with respect to x

4. Differentiate the inner function to get (d(9)*x)/d(x)=9

5. Substitute these values into the chain rule formula.

d(ln(9*x))/d(x)=1/(9*x)⋅9

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

9/(9*x)=1/x

Final Answer

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

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