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

Problem

d()/d(x)*(e^x−e^(−x))

Solution

1. Apply the sum/difference rule of differentiation, which allows for the derivative of each term to be taken separately.

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

2. Differentiate the first term using the rule d(e^x)/d(x)=e^x

d(e^x)/d(x)=e^x

3. Differentiate the second term using the chain rule, where the derivative of e^u is e^u⋅d(u)/d(x)

d(e^(−x))/d(x)=e^(−x)⋅d(−x)/d(x)

4. Calculate the derivative of the exponent −x which is −1

d(e^(−x))/d(x)=e^(−x)⋅(−1)

5. Simplify the expression by substituting the results back into the difference rule and combining the signs.

e^x−(−e^(−x))=e^x+e^(−x)

Final Answer

d()/d(x)*(e^x−e^(−x))=e^x+e^(−x)

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