Find the Derivative - d/dx y=(e^x)/(1-e^x)

Problem

d()/d(x)(ex)/(1−ex)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two terms, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the numerator and denominator. Let u=ex and v=1−ex

3. Differentiate the individual components.

d(ex)/d(x)=ex

d(1−ex)/d(x)=−ex

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=((1−ex)*(ex)−(ex)*(−ex))/((1−ex)2)

5. Distribute the terms in the numerator.

d(y)/d(x)=(ex−e(2*x)+e(2*x))/((1−ex)2)

6. Simplify the numerator by combining like terms. The −e(2*x) and +e(2*x) cancel out.

d(y)/d(x)=(ex)/((1−ex)2)

Final Answer

d()/d(x)(ex)/(1−ex)=(ex)/((1−ex)2)

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