Find the Derivative - d/d@VAR f(x)=(e^x)/(3+e^x)

Problem

d()/d(x)(ex)/(3+ex)

Solution

1. Identify the function as a quotient of two differentiable functions, u=ex and v=3+ex

2. Apply the quotient rule, which states that d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator and denominator: d(ex)/d(x)=ex and d(3+ex)/d(x)=ex

4. Substitute these derivatives into the quotient rule formula.

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

5. Distribute the ex in the numerator.

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

6. Simplify the numerator by canceling the e(2*x) terms.

d()/d(x)(ex)/(3+ex)=(3*ex)/((3+ex)2)

Final Answer

d()/d(x)(ex)/(3+ex)=(3*ex)/((3+ex)2)

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