What is the derivative of ((sin(x)/(cos(2x)))^3) ?

The derivative of ((sin(x)/(cos(2x)))^3) is 3sin^2(x)cos(x)sec^3(2x)+6sec^3(2x)tan(2x)sin^3(x)

What is the first derivative of ((sin(x)/(cos(2x)))^3) ?

The first derivative of ((sin(x)/(cos(2x)))^3) is 3sin^2(x)cos(x)sec^3(2x)+6sec^3(2x)tan(2x)sin^3(x)

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

3 sin^{2} (x) cos (x) sec^{3} (2 x) + 6 sec^{3} (2 x) tan (2 x) sin^{3} (x)

Solution steps

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

Simplify

(\frac{sin ( x )}{cos ( 2 x )})^{3} : sin^{3} (x) sec^{3} (2 x)

= \frac{d}{d x} (sin^{3} (x) sec^{3} (2 x))

Apply the Product Rule:

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

f = sin^{3} (x), g = sec^{3} (2 x)

= \frac{d}{d x} (sin^{3} (x)) sec^{3} (2 x) + \frac{d}{d x} (sec^{3} (2 x)) sin^{3} (x)

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

\frac{d}{d x} (sec^{3} (2 x)) = 6 sec^{3} (2 x) tan (2 x)

= 3 sin^{2} (x) cos (x) sec^{3} (2 x) + 6 sec^{3} (2 x) tan (2 x) sin^{3} (x)

What is the derivative of ((sin(x)/(cos(2x)))^3) ?
The derivative of ((sin(x)/(cos(2x)))^3) is 3sin^2(x)cos(x)sec^3(2x)+6sec^3(2x)tan(2x)sin^3(x)
What is the first derivative of ((sin(x)/(cos(2x)))^3) ?
The first derivative of ((sin(x)/(cos(2x)))^3) is 3sin^2(x)cos(x)sec^3(2x)+6sec^3(2x)tan(2x)sin^3(x)