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Popular Calculus >

derivative of xe^{-x}cos(x)

Solution

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

Solution

e^{- x} cos (x) + x (- e^{- x} cos (x) - e^{- x} sin (x))

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

Graph

Popular Examples

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

limit as x approaches 0 of sin(x/x)

x \to 0 lim (sin (\frac{x}{x}))

derivative of-8x^2

\frac{d}{d x} (- 8 x^{2})

(6x^2y^2-4y^4)dx+(2x^3y-4xy^3)dy=0

(6 x^{2} y^{2} - 4 y^{4}) d x + (2 x^{3} y - 4 x y^{3}) d y = 0

integral of cos(pix-npix)

\int cos (π x - nπ x) d x

Frequently Asked Questions (FAQ)

What is the derivative of xe^{-x}cos(x) ?
The derivative of xe^{-x}cos(x) is e^{-x}cos(x)+x(-e^{-x}cos(x)-e^{-x}sin(x))
What is the first derivative of xe^{-x}cos(x) ?
The first derivative of xe^{-x}cos(x) is e^{-x}cos(x)+x(-e^{-x}cos(x)-e^{-x}sin(x))