Find the Derivative - d/dx 1/2*(e^x+e^(-x))

Problem

d()/d(x)1/2*(e^x+e^(−x))

Solution

1. Apply the constant multiple rule to move the constant factor 1/2 outside of the derivative.

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

2. Apply the sum rule to differentiate each term inside the parentheses individually.

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

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

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

4. Differentiate the second term using the chain rule, where d(e^u)/d(x)=e^ud(u)/d(x) with u=−x

d(e^(−x))/d(x)=e^(−x)⋅(−1)=−e^(−x)

5. Combine the results back into the expression and simplify.

1/2*(e^x−e^(−x))

Final Answer

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

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