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

The derivative of (e^{-x}dx) is d(-e^{-x}x+e^{-x})

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

The first derivative of (e^{-x}dx) is d(-e^{-x}x+e^{-x})

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

d (- e^{- x} x + e^{- x})

Solution steps

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

Treat

d

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

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

\frac{d x}{d x} = 1

= d ((- e^{- x}) x + 1 \cdot e^{- x})

Simplify

= d (- e^{- x} x + e^{- x})

\frac{d}{d x} (- e^{- 1})

x \to 4 lim (\frac{3}{4 - x})

\int_{- 2}^{3} \frac{7}{x ^{4}} d x

\int (2 x - 3)^{- 7} d x

\int \frac{8}{x + 1} + x d x

What is the derivative of (e^{-x}dx) ?
The derivative of (e^{-x}dx) is d(-e^{-x}x+e^{-x})
What is the first derivative of (e^{-x}dx) ?
The first derivative of (e^{-x}dx) is d(-e^{-x}x+e^{-x})