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

Problem

d()/d(x)(e(−x)+1)/(ex)

Solution

1. Simplify the expression by dividing each term in the numerator by the denominator to avoid the quotient rule.

(e(−x)+1)/(ex)=(e(−x))/(ex)+1/(ex)

2. Apply the laws of exponents to rewrite the terms as powers of e

(e(−x))/(ex)+1/(ex)=e(−2*x)+e(−x)

3. Differentiate the expression term by term using the chain rule, where d(eu)/d(x)=eud(u)/d(x)

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

4. Calculate the derivative of each term.

d(e(−2*x))/d(x)=−2*e(−2*x)

d(e(−x))/d(x)=−e(−x)

5. Combine the results into a single expression.

−2*e(−2*x)−e(−x)

Final Answer

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

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