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

Problem

d()/d(x)(ex+e(−x))/2

Solution

1. Identify the constant factor in the expression and move it outside the derivative operator.

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

2. Apply the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives.

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

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

d(ex)/d(x)=ex

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

d(e(−x))/d(x)=e(−x)⋅(−1)=−e(−x)

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

1/2*(ex−e(−x))=(ex−e(−x))/2

Final Answer

d()/d(x)(ex+e(−x))/2=(ex−e(−x))/2

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