Find the Derivative - d/dx natural log of x^3

Problem

d(ln(x^3))/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 chain rule for the natural logarithm, where d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Substitute u=x^3 into the formula.

d(ln(x^3))/d(x)=1/(x^3)⋅d(x^3)/d(x)

4. Differentiate the inner function x^3 using the power rule.

d(x^3)/d(x)=3*x^2

5. Multiply the results together.

d(ln(x^3))/d(x)=1/(x^3)⋅3*x^2

6. Simplify the expression by canceling the common x^2 terms.

(3*x^2)/(x^3)=3/x

Final Answer

d(ln(x^3))/d(x)=3/x

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