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

Problem

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

Solution

1. Apply the property of logarithms to simplify the expression before differentiating. The power rule for logarithms states ln(ab)=b*ln(a)

ln(x5)=5*ln(x)

2. Apply the constant multiple rule for derivatives, which allows the constant 5 to be moved outside the derivative.

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

3. Differentiate the natural log function using the rule d(ln(x))/d(x)=1/x

5⋅1/x

4. Simplify the resulting expression by multiplying the constant and the fraction.

5/x

Final Answer

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

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