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

The derivative of e^{-x}ln(x) is -e^{-x}ln(x)+(e^{-x})/x

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

The first derivative of e^{-x}ln(x) is -e^{-x}ln(x)+(e^{-x})/x

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

- e^{- x} ln (x) + \frac{e ^{- x}}{x}

Solution steps

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

Apply the Product Rule:

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

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

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

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

\frac{d}{d x} (ln (x)) = \frac{1}{x}

= (- e^{- x}) ln (x) + \frac{1}{x} e^{- x}

Simplify

(- e^{- x}) ln (x) + \frac{1}{x} e^{- x} : - e^{- x} ln (x) + \frac{e ^{- x}}{x}

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

\int_{0}^{1} - x^{2} + 4 x + 1 d x

\int \frac{1}{7 x + 4} d x

\frac{d y}{d x}, x y - 6 = 0

\int 2 sin^{3} (5 x) cos^{3} (5 x) d x

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