What is the derivative of sin(2xcsc(x/2)) ?

The derivative of sin(2xcsc(x/2)) is 2cos(2x)csc(x/2)-1/2 cot(x/2)csc(x/2)sin(2x)

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

The first derivative of sin(2xcsc(x/2)) is 2cos(2x)csc(x/2)-1/2 cot(x/2)csc(x/2)sin(2x)

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

2 cos (2 x) csc (\frac{x}{2}) - \frac{1}{2} cot (\frac{x}{2}) csc (\frac{x}{2}) sin (2 x)

Solution steps

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

Apply the Product Rule:

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

f = sin (2 x), g = csc (\frac{x}{2})

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

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

\frac{d}{d x} (csc (\frac{x}{2})) = - cot (\frac{x}{2}) csc (\frac{x}{2}) \frac{1}{2}

= cos (2 x) \cdot 2 csc (\frac{x}{2}) + (- cot (\frac{x}{2}) csc (\frac{x}{2}) \frac{1}{2}) sin (2 x)

Simplify

= 2 cos (2 x) csc (\frac{x}{2}) - \frac{1}{2} cot (\frac{x}{2}) csc (\frac{x}{2}) sin (2 x)

\int 8 sin (- 2 x + 7) d x

x \to 3 + lim (x - 3)

x \to 1 lim (2 x - 1)

y = x^{3}, y = x^{2} + x

\int 4 x + \frac{1}{x} d x

What is the derivative of sin(2xcsc(x/2)) ?
The derivative of sin(2xcsc(x/2)) is 2cos(2x)csc(x/2)-1/2 cot(x/2)csc(x/2)sin(2x)
What is the first derivative of sin(2xcsc(x/2)) ?
The first derivative of sin(2xcsc(x/2)) is 2cos(2x)csc(x/2)-1/2 cot(x/2)csc(x/2)sin(2x)