Find the Derivative - d/dx natural log of 5x

Problem

d(ln(5*x))/d(x)

Solution

1. Identify the function as a composition of the natural logarithm function 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 the inner function u=5*x and find its derivative with respect to x

4. Differentiate the inner function to get (d(5)*x)/d(x)=5

5. Substitute these values into the Chain Rule formula to get 1/(5*x)⋅5

6. Simplify the expression by canceling the common factor of 5 in the numerator and denominator.

Final Answer

d(ln(5*x))/d(x)=1/x

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