Find the Derivative - d/dx (3-xe^x)/(x+e^x)

Problem

d()/d(x)(3−x*e^x)/(x+e^x)

Solution

1. Identify the rule needed for differentiation, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the numerator and denominator functions: u=3−x*e^x and v=x+e^x

3. Differentiate the numerator u using the product rule for the term x*e^x d(u)/d(x)=0−(xd(e^x)/d(x)+e^xd(x)/d(x))=−(x*e^x+e^x)

4. Differentiate the denominator v d(v)/d(x)=1+e^x

5. Substitute these components into the quotient rule formula: ((x+e^x)*(−(x*e^x+e^x))−(3−x*e^x)*(1+e^x))/((x+e^x)^2)

6. Expand the terms in the numerator: −(x^2*e^x+x*e^x+x*e^(2*x)+e^(2*x))−(3+3*e^x−x*e^x−x*e^(2*x))

7. Distribute the negative signs: −x^2*e^x−x*e^x−x*e^(2*x)−e^(2*x)−3−3*e^x+x*e^x+x*e^(2*x)

8. Simplify the numerator by canceling the terms −x*e^x and +x*e^x and −x*e^(2*x) and +x*e^(2*x) −x^2*e^x−e^(2*x)−3*e^x−3

9. Factor out a negative sign from the numerator to reach the final form: −(x^2*e^x+e^(2*x)+3*e^x+3)

Final Answer

d()/d(x)(3−x*e^x)/(x+e^x)=(−(x^2*e^x+e^(2*x)+3*e^x+3))/((x+e^x)^2)

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