Question
Find the derivative of the function. \( f(x)=\arctan \left(e^{x}\right) \) \( f^{\prime}(x)=\square \)
Ask by Ramos Griffin. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f^{\prime}(x) = \frac{e^{x}}{1 + e^{2x}} \)
Solution
To differentiate f(x) = arctan(eˣ), we use the chain rule. The derivative of arctan(u) with respect to u is 1/(1 + u²), and the derivative of u = eˣ is eˣ.
Thus,
f '(x) = (1/(1 + (eˣ)²)) · eˣ = eˣ/(1 + e^(2x)).
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Simplify this solution Mind Expander
To find the derivative of the function \( f(x) = \arctan(e^{x}) \), we can apply the chain rule. The derivative of \( \arctan(u) \) is \( \frac{1}{1+u^2} \), where \( u = e^{x} \). Thus, the derivative of \( f(x) \) is: \[ f'(x) = \frac{1}{1 + (e^{x})^2} \cdot \frac{d}{dx}(e^{x}) = \frac{1}{1 + e^{2x}} \cdot e^{x} \] So, we have: \[ f'(x) = \frac{e^{x}}{1 + e^{2x}} \]
