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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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

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

\frac{( x ^{2} + 32 x )}{8}

y = 1 - \frac{x ^{2}}{36}

\frac{\partial}{\partial z} (2 y + z)

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