What is the derivative of sin(2xcos(2x)) ?

The derivative of sin(2xcos(2x)) is 2cos(4x)

What is the first derivative of sin(2xcos(2x)) ?

The first derivative of sin(2xcos(2x)) is 2cos(4x)

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

2 cos (4 x)

Solution steps

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

Apply the Product Rule:

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

f = sin (2 x), g = cos (2 x)

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

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

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

= cos (2 x) \cdot 2 cos (2 x) + (- sin (2 x) \cdot 2) sin (2 x)

Simplify

cos (2 x) \cdot 2 cos (2 x) + (- sin (2 x) \cdot 2) sin (2 x) : 2 cos (4 x)

= 2 cos (4 x)

What is the derivative of sin(2xcos(2x)) ?
The derivative of sin(2xcos(2x)) is 2cos(4x)
What is the first derivative of sin(2xcos(2x)) ?
The first derivative of sin(2xcos(2x)) is 2cos(4x)