Find the Derivative - d/dx y = natural log of x^2e^x

Problem

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

Solution

1. Apply logarithm properties to simplify the expression before differentiating. Use the product rule for logarithms ln(a*b)=ln(a)+ln(b)

ln(x2*ex)=ln(x2)+ln(ex)

2. Further simplify using the power rule for logarithms ln(ab)=b*ln(a) and the fact that ln(ex)=x

ln(x2)+ln(ex)=2*ln(x)+x

3. Differentiate the simplified expression term by term. Apply the rule d(ln(x))/d(x)=1/x and the power rule for x

d()/d(x)*(2*ln(x)+x)=2⋅1/x+1

4. Combine the terms into a single expression.

2/x+1=(2+x)/x

Final Answer

d(ln(x2*ex))/d(x)=(2+x)/x

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