Find the Derivative - d/dx natural log of e^x+xe^x

Problem

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

Solution

1. Simplify the expression inside the natural logarithm by factoring out the common term ex

ln(ex+x*ex)=ln(ex*(1+x))

2. Apply the logarithmic property ln(a*b)=ln(a)+ln(b) to separate the terms.

ln(ex*(1+x))=ln(ex)+ln(1+x)

3. Simplify the first term using the identity ln(ex)=x

x+ln(1+x)

4. Differentiate the simplified expression term by term with respect to x

d(x)/d(x)+d(ln(1+x))/d(x)

5. Apply the chain rule to the second term, where the derivative of ln(u) is 1/u⋅d(u)/d(x)

1+1/(1+x)⋅d(1+x)/d(x)

6. Evaluate the derivative of 1 + x,w*h*i*c*h*i*s()$.

1+1/(1+x)

7. Combine the terms into a single fraction by finding a common denominator.

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

Final Answer

d(ln(ex+x*ex))/d(x)=(x+2)/(x+1)

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