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

Problem

d(ln(x5))/d(x)

Solution

1. Identify the function as a composition of the natural logarithm and a power function, which requires the use of the chain rule or logarithmic properties.

2. Apply the logarithmic power rule, ln(ab)=b*ln(a) to simplify the expression before differentiating.

ln(x5)=5*ln(x)

3. Differentiate the simplified expression using the constant multiple rule and the derivative of the natural logarithm, d(ln(x))/d(x)=1/x

(d(5)*ln(x))/d(x)=5⋅1/x

4. Simplify the resulting expression into a single fraction.

5⋅1/x=5/x

Final Answer

d(ln(x5))/d(x)=5/x

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