Question
Find the derivative of the function. \[ f(x)=\arctan \left(e^{x}\right) \] \( f^{\prime}(x)=\square \)
Ask by Fowler Pollard. in the United States
Feb 26,2025
Upstudy AI Solution
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Answer
The derivative of \( f(x) = \arctan(e^x) \) is \( f'(x) = \frac{e^x}{1 + e^{2x}} \).
Solution
To find the derivative of f(x) = arctan(e^x), we can use the chain rule. The derivative of arctan(u) with respect to u is 1/(1+u²), and the derivative of e^x with respect to x is e^x.
Let u = e^x. Then:
f'(x) = (1/(1+u²)) · (du/dx) = (1/(1+e^(2x))) · e^x.
Thus, the derivative is:
f'(x) = e^x/(1+e^(2x)).
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Simplify this solution The Deep Dive
To find the derivative of the function \( f(x) = \arctan(e^{x}) \), we can use the chain rule. The derivative of \( \arctan(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \), where \( u = e^{x} \). Therefore, applying the chain rule, we get: \[ f'(x) = \frac{1}{1 + (e^{x})^2} \cdot e^{x} \] Thus, the derivative is: \[ f'(x) = \frac{e^{x}}{1 + e^{2x}} \]
