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

Problem

d(arctan(ex))/d(x)

Solution

1. Identify the outer function as arctan(u) and the inner function as u=ex

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+u2)

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

5. Substitute these components into the chain rule formula:

d(arctan(ex))/d(x)=1/(1+(ex)2)⋅ex

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

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

Final Answer

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

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