What is the derivative of ue^{-x} ?

The derivative of ue^{-x} is e^{-x}(du)/(dx)-e^{-x}u

What is the first derivative of ue^{-x} ?

The first derivative of ue^{-x} is e^{-x}(du)/(dx)-e^{-x}u

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

e^{- x} \frac{d u}{d x} - e^{- x} u

Solution steps

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

Apply the Product Rule:

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

f = u, g = e^{- x}

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

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

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

Simplify

= e^{- x} \frac{d u}{d x} - e^{- x} u

\int 361 - x^{2} d x

\int - 2 x^{- 1} d x

\int (\frac{e ^{x}}{3} + 2 x) d x

\int tan^{5} (x) \cdot sec^{3} (x) d x

\frac{d y}{d x} = - 4^{- 2} - y^{- 1} + y^{- 1} + y

What is the derivative of ue^{-x} ?
The derivative of ue^{-x} is e^{-x}(du)/(dx)-e^{-x}u
What is the first derivative of ue^{-x} ?
The first derivative of ue^{-x} is e^{-x}(du)/(dx)-e^{-x}u