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

Problem

d()/d(x)(e^x−e^(−x))/(e^x+e^(−x))

Solution

1. Identify the function as a quotient of two terms, which can be simplified by recognizing the hyperbolic tangent function tanh(x)=(e^x−e^(−x))/(e^x+e^(−x))

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

3. Differentiate the numerator u=e^x−e^(−x) to get d(u)/d(x)=e^x+e^(−x)

4. Differentiate the denominator v=e^x+e^(−x) to get d(v)/d(x)=e^x−e^(−x)

5. Substitute these derivatives into the quotient rule formula:

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

6. Simplify the numerator by expanding the squares:

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

7. Combine like terms in the numerator to find the final simplified form:

d()/d(x)(e^x−e^(−x))/(e^x+e^(−x))=4/((e^x+e^(−x))^2)

Final Answer

d()/d(x)(e^x−e^(−x))/(e^x+e^(−x))=4/((e^x+e^(−x))^2)

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