Find the Derivative Using Quotient Rule - d/dc (1+ce^x)/(1-ce^x)

Problem

d()/d(c)(1+c*e^x)/(1−c*e^x)

Solution

1. Identify the numerator u=1+c*e^x and the denominator v=1−c*e^x for the quotient rule.

2. Differentiate u and v with respect to c treating x as a constant.

3. Calculate the derivative of the numerator: d(u)/d(c)=e^x

4. Calculate the derivative of the denominator: d(v)/d(c)=−e^x

5. Apply the quotient rule formula d()/d(c)u/v=(vd(u)/d(c)−ud(v)/d(c))/(v^2)

6. Substitute the expressions into the formula: ((1−c*e^x)*(e^x)−(1+c*e^x)*(−e^x))/((1−c*e^x)^2)

7. Simplify the numerator by distributing the terms: e^x−c*e^(2*x)+e^x+c*e^(2*x)

8. Combine like terms in the numerator to get 2*e^x

Final Answer

d()/d(c)(1+c*e^x)/(1−c*e^x)=(2*e^x)/((1−c*e^x)^2)

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