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

Problem

d()/d(x)*ln(5*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 for the natural logarithm, which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Substitute u=5*x and find its derivative.

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

4. Combine the components to find the derivative of the entire expression.

d(ln(5*x))/d(x)=1/(5*x)⋅5

5. Simplify the resulting fraction by canceling the common factor of 5.

5/(5*x)=1/x

Final Answer

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

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