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

Problem

d()/d(x)(3−x*ex)/(x+ex)

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))/(v2)

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

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

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

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

6. Expand the terms in the numerator: −(x2*ex+x*ex+x*e(2*x)+e(2*x))−(3+3*ex−x*ex−x*e(2*x))

7. Distribute the negative signs: −x2*ex−x*ex−x*e(2*x)−e(2*x)−3−3*ex+x*ex+x*e(2*x)

8. Simplify the numerator by canceling the terms −x*ex and +x*ex and −x*e(2*x) and +x*e(2*x) −x2*ex−e(2*x)−3*ex−3

9. Factor out a negative sign from the numerator to reach the final form: −(x2*ex+e(2*x)+3*ex+3)

Final Answer

d()/d(x)(3−x*ex)/(x+ex)=(−(x2*ex+e(2*x)+3*ex+3))/((x+ex)2)

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