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

Problem

d()/d(x)(e^x−1)/(e^x+1)

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the numerator and denominator functions: u=e^x−1 and v=e^x+1

3. Differentiate both parts with respect to x d(e^x−1)/d(x)=e^x and d(e^x+1)/d(x)=e^x

4. Substitute these into the quotient rule formula:

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

5. Distribute the e^x in the numerator:

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

6. Simplify the numerator by combining like terms:

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

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

Final Answer

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

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