Find the Derivative - d/d@VAR f(x) = 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.

2. Apply the power rule for logarithms to simplify the expression before differentiating by moving the exponent in front of the logarithm.

ln(x5)=5*ln(x)

3. Apply the constant multiple rule to move the constant factor outside of the derivative.

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

4. Differentiate the natural logarithm function using the rule d(ln(x))/d(x)=1/x

5⋅1/x=5/x

Final Answer

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

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