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

Problem

d()/d(x)(e^x)/(3+e^x)

Solution

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

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

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

4. Substitute these derivatives into the quotient rule formula.

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

5. Distribute the e^x in the numerator.

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

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

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

Final Answer

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

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