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

Problem

d(arctan(e^x))/d(x)

Solution

1. Identify the outer function as arctan(u) and the inner function as u=e^x

2. Apply the chain rule, which states that (d(ƒ)*(g(x)))/d(x)=ƒ^′*(g(x))⋅g(x)^′

3. Recall the derivative of the arctangent function: d(arctan(u))/d(u)=1/(1+u^2)

4. Differentiate the inner function u=e^x to get d(e^x)/d(x)=e^x

5. Substitute these components into the chain rule formula:

d(arctan(e^x))/d(x)=1/(1+(e^x)^2)⋅e^x

6. Simplify the expression by using the power of a power rule (e^x)^2=e^(2*x)

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

Final Answer

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

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