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

Problem

d(arctan(e(2*x)))/d(x)

Solution

1. Identify the outer function as arctan(u) and the inner function as u=e(2*x)

2. Apply the chain rule for the derivative of the arctangent function, which states d(arctan(u))/d(x)=1/(1+u2)⋅d(u)/d(x)

3. Differentiate the inner function u=e(2*x) using the chain rule again, which gives d(e(2*x))/d(x)=e(2*x)⋅2

4. Substitute these components back into the chain rule formula.

5. Simplify the expression by multiplying the terms and applying the power of a power rule (e(2*x))2=e(4*x)

Final Answer

d(arctan(e(2*x)))/d(x)=(2*e(2*x))/(1+e(4*x))

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