What is the derivative of e^{-3x}(2cos(2x-3sin(2x))) ?

The derivative of e^{-3x}(2cos(2x-3sin(2x))) is -12e^{-3x}cos(2x)+5e^{-3x}sin(2x)

What is the first derivative of e^{-3x}(2cos(2x-3sin(2x))) ?

The first derivative of e^{-3x}(2cos(2x-3sin(2x))) is -12e^{-3x}cos(2x)+5e^{-3x}sin(2x)

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\frac{d}{d x} (e^{- 3 x} (2 cos (2 x) - 3 sin (2 x)))

- 12 e^{- 3 x} cos (2 x) + 5 e^{- 3 x} sin (2 x)

Solution steps

\frac{d}{d x} (e^{- 3 x} (2 cos (2 x) - 3 sin (2 x)))

Apply the Product Rule:

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

f = e^{- 3 x}, g = 2 cos (2 x) - 3 sin (2 x)

= \frac{d}{d x} (e^{- 3 x}) (2 cos (2 x) - 3 sin (2 x)) + \frac{d}{d x} (2 cos (2 x) - 3 sin (2 x)) e^{- 3 x}

\frac{d}{d x} (e^{- 3 x}) = - 3 e^{- 3 x}

\frac{d}{d x} (2 cos (2 x) - 3 sin (2 x)) = - 4 sin (2 x) - 6 cos (2 x)

= (- 3 e^{- 3 x}) (2 cos (2 x) - 3 sin (2 x)) + (- 4 sin (2 x) - 6 cos (2 x)) e^{- 3 x}

Simplify

(- 3 e^{- 3 x}) (2 cos (2 x) - 3 sin (2 x)) + (- 4 sin (2 x) - 6 cos (2 x)) e^{- 3 x} : - 12 e^{- 3 x} cos (2 x) + 5 e^{- 3 x} sin (2 x)

= - 12 e^{- 3 x} cos (2 x) + 5 e^{- 3 x} sin (2 x)

x \to 0 lim (\frac{2}{3 - e ^{\frac{1}{x}}})

\int \frac{1}{x ( 1 - x )} d x

\int (x^{2} - x) e^{x} d x

y^{^{'}} = y (1 - y), y (0) = 12

What is the derivative of e^{-3x}(2cos(2x-3sin(2x))) ?
The derivative of e^{-3x}(2cos(2x-3sin(2x))) is -12e^{-3x}cos(2x)+5e^{-3x}sin(2x)
What is the first derivative of e^{-3x}(2cos(2x-3sin(2x))) ?
The first derivative of e^{-3x}(2cos(2x-3sin(2x))) is -12e^{-3x}cos(2x)+5e^{-3x}sin(2x)