Find the Derivative - d/dx natural log of 3x^2

Problem

d(ln(3*x2))/d(x)

Solution

1. Identify the outer function as ln(u) and the inner function as u=3*x2

2. Apply the chain rule, which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Differentiate the inner function 3*x2 with respect to x using the power rule to get 6*x

4. Substitute the expressions back into the chain rule formula.

d(ln(3*x2))/d(x)=1/(3*x2)⋅6*x

5. Simplify the resulting fraction by canceling common factors.

(6*x)/(3*x2)=2/x

Final Answer

d(ln(3*x2))/d(x)=2/x

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