Find the Derivative - d/dx natural log of ex

Problem

d(ln(e*x))/d(x)

Solution

1. Apply the property of logarithms to simplify the expression before differentiating. The natural log of a product is the sum of the logs: ln(e*x)=ln(e)+ln(x)

2. Simplify the constant term using the fact that ln(e)=1 The expression becomes 1+ln(x)

3. Apply the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives: d(1+ln(x))/d(x)=d(1)/d(x)+d(ln(x))/d(x)

4. Differentiate each term individually. The derivative of a constant 1 is 0 and the derivative of ln(x) is 1/x

5. Combine the results to find the final derivative: 0+1/x=1/x

Final Answer

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

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