What is (\partial)/(\partial x)(tan(2x)sin(2x)) ?

The answer to (\partial)/(\partial x)(tan(2x)sin(2x)) is 2sin(2x)+2sec(2x)tan(2x)

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(\partial)/(\partial x)(tan(2x)sin(2x))

\frac{\partial}{\partial x} (tan (2 x) sin (2 x))

2 sin (2 x) + 2 sec (2 x) tan (2 x)

Solution steps

\frac{\partial}{\partial x} (tan (2 x) sin (2 x))

Apply the Product Rule:

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

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

= \frac{\partial}{\partial x} (tan (2 x)) sin (2 x) + \frac{\partial}{\partial x} (sin (2 x)) tan (2 x)

\frac{\partial}{\partial x} (tan (2 x)) = sec^{2} (2 x) \cdot 2

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

= sec^{2} (2 x) \cdot 2 sin (2 x) + cos (2 x) \cdot 2 tan (2 x)

Simplify

sec^{2} (2 x) \cdot 2 sin (2 x) + cos (2 x) \cdot 2 tan (2 x) : 2 sin (2 x) + 2 sec (2 x) tan (2 x)

= 2 sin (2 x) + 2 sec (2 x) tan (2 x)

What is (\partial)/(\partial x)(tan(2x)sin(2x)) ?
The answer to (\partial)/(\partial x)(tan(2x)sin(2x)) is 2sin(2x)+2sec(2x)tan(2x)