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

Problem

d()/d(x)(ex−e(−x))/(ex+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)=(ex−e(−x))/(ex+e(−x))

2. Apply the quotient rule for differentiation, which states d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

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

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

5. Substitute these derivatives into the quotient rule formula:

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

6. Simplify the numerator by expanding the squares:

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

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

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

Final Answer

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

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