What is the derivative of e^{-x}arctan(2e^x) ?

The derivative of e^{-x}arctan(2e^x) is -e^{-x}arctan(2e^x)+2/(4e^{2x)+1}

What is the first derivative of e^{-x}arctan(2e^x) ?

The first derivative of e^{-x}arctan(2e^x) is -e^{-x}arctan(2e^x)+2/(4e^{2x)+1}

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derivative of

e^{- x} arctan (2 e^{x})

- e^{- x} arctan (2 e^{x}) + \frac{2}{4 e ^{2 x} + 1}

Solution steps

\frac{d}{d x} (e^{- x} arctan (2 e^{x}))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- x}, g = arctan (2 e^{x})

= \frac{d}{d x} (e^{- x}) arctan (2 e^{x}) + \frac{d}{d x} (arctan (2 e^{x})) e^{- x}

\frac{d}{d x} (e^{- x}) = - e^{- x}

\frac{d}{d x} (arctan (2 e^{x})) = \frac{2 e ^{x}}{4 e ^{2 x} + 1}

= (- e^{- x}) arctan (2 e^{x}) + \frac{2 e ^{x}}{4 e ^{2 x} + 1} e^{- x}

Simplify

(- e^{- x}) arctan (2 e^{x}) + \frac{2 e ^{x}}{4 e ^{2 x} + 1} e^{- x} : - e^{- x} arctan (2 e^{x}) + \frac{2}{4 e ^{2 x} + 1}

= - e^{- x} arctan (2 e^{x}) + \frac{2}{4 e ^{2 x} + 1}

\int (3 x^{2} + 4 x + 1) d x

n = 0 \sum \infty (- x^{2} - 8)^{n}

\int \frac{1}{2} e^{x^{2}} d x

(x^{2} - 4) \cdot \frac{d y}{d x} + 4 y = (x + 2)^{2}

\int 12 x y d x

What is the derivative of e^{-x}arctan(2e^x) ?
The derivative of e^{-x}arctan(2e^x) is -e^{-x}arctan(2e^x)+2/(4e^{2x)+1}
What is the first derivative of e^{-x}arctan(2e^x) ?
The first derivative of e^{-x}arctan(2e^x) is -e^{-x}arctan(2e^x)+2/(4e^{2x)+1}