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

The derivative of e^{-x}cos(sqrt(x)) is -e^{-x}cos(sqrt(x))-(e^{-x}sin(sqrt(x)))/(2sqrt(x))

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

The first derivative of e^{-x}cos(sqrt(x)) is -e^{-x}cos(sqrt(x))-(e^{-x}sin(sqrt(x)))/(2sqrt(x))

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

- e^{- x} cos (x) - \frac{e ^{- x} sin ( x )}{2 x}

Solution steps

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

Apply the Product Rule:

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

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

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

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

\frac{d}{d x} (cos (x)) = - \frac{sin ( x )}{2 x}

= (- e^{- x}) cos (x) + (- \frac{sin ( x )}{2 x}) e^{- x}

Simplify

(- e^{- x}) cos (x) + (- \frac{sin ( x )}{2 x}) e^{- x} : - e^{- x} cos (x) - \frac{e ^{- x} sin ( x )}{2 x}

= - e^{- x} cos (x) - \frac{e ^{- x} sin ( x )}{2 x}

What is the derivative of e^{-x}cos(sqrt(x)) ?
The derivative of e^{-x}cos(sqrt(x)) is -e^{-x}cos(sqrt(x))-(e^{-x}sin(sqrt(x)))/(2sqrt(x))
What is the first derivative of e^{-x}cos(sqrt(x)) ?
The first derivative of e^{-x}cos(sqrt(x)) is -e^{-x}cos(sqrt(x))-(e^{-x}sin(sqrt(x)))/(2sqrt(x))