What is the derivative of y=e^{-x}arctan(5e^x) ?

The derivative of y=e^{-x}arctan(5e^x) is -e^{-x}arctan(5e^x)+5/(25e^{2x)+1}

What is the first derivative of y=e^{-x}arctan(5e^x) ?

The first derivative of y=e^{-x}arctan(5e^x) is -e^{-x}arctan(5e^x)+5/(25e^{2x)+1}

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derivative of

y = e^{- x} arctan (5 e^{x})

- e^{- x} arctan (5 e^{x}) + \frac{5}{25 e ^{2 x} + 1}

Solution steps

\frac{d}{d x} (e^{- x} arctan (5 e^{x}))

Apply the Product Rule:

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

f = e^{- x}, g = arctan (5 e^{x})

= \frac{d}{d x} (e^{- x}) arctan (5 e^{x}) + \frac{d}{d x} (arctan (5 e^{x})) e^{- x}

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

\frac{d}{d x} (arctan (5 e^{x})) = \frac{5 e ^{x}}{25 e ^{2 x} + 1}

= (- e^{- x}) arctan (5 e^{x}) + \frac{5 e ^{x}}{25 e ^{2 x} + 1} e^{- x}

Simplify

(- e^{- x}) arctan (5 e^{x}) + \frac{5 e ^{x}}{25 e ^{2 x} + 1} e^{- x} : - e^{- x} arctan (5 e^{x}) + \frac{5}{25 e ^{2 x} + 1}

= - e^{- x} arctan (5 e^{x}) + \frac{5}{25 e ^{2 x} + 1}

x^{2} ln (x^{2} + 1)

\frac{d y}{d x}, y^{3} = x^{8}

\int \frac{( x + 6 ) ^{4}}{2 x} d x

sin (- 1 (4 x + 1))

\int_{0}^{\frac{π}{2}} sin^{2} (t) d t

What is the derivative of y=e^{-x}arctan(5e^x) ?
The derivative of y=e^{-x}arctan(5e^x) is -e^{-x}arctan(5e^x)+5/(25e^{2x)+1}
What is the first derivative of y=e^{-x}arctan(5e^x) ?
The first derivative of y=e^{-x}arctan(5e^x) is -e^{-x}arctan(5e^x)+5/(25e^{2x)+1}