What is (\partial)/(\partial x)(x*arccsc(x/2)) ?

The answer to (\partial)/(\partial x)(x*arccsc(x/2)) is arccsc(x/2)-2/(sqrt(x^2-4))

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(\partial)/(\partial x)(x*arccsc(x/2))

\frac{\partial}{\partial x} (x \cdot arccsc (\frac{x}{2}))

arccsc (\frac{x}{2}) - \frac{2}{x ^{2} - 4}

Solution steps

\frac{\partial}{\partial x} (x arccsc (\frac{x}{2}))

Apply the Product Rule:

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

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

= \frac{\partial}{\partial x} (x) arccsc (\frac{x}{2}) + \frac{\partial}{\partial x} (arccsc (\frac{x}{2})) x

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (arccsc (\frac{x}{2})) = - \frac{2}{x x ^{2} - 4}

= 1 \cdot arccsc (\frac{x}{2}) + (- \frac{2}{x x ^{2} - 4}) x

Simplify

1 \cdot arccsc (\frac{x}{2}) + (- \frac{2}{x x ^{2} - 4}) x : arccsc (\frac{x}{2}) - \frac{2}{x ^{2} - 4}

= arccsc (\frac{x}{2}) - \frac{2}{x ^{2} - 4}

\int 5 x^{2} + 1 d x

\int \frac{a e ^{x} + b}{a e ^{x} - b} d x

(9 x y^{2} - 4 y) + (9 x^{2} y - 4 x) y^{^{'}} = 0

\int (csc (3 x) - cot (3 x))^{2} d x

y^{^{''}} - 2 y^{^{'}} = sin (x), y (0) = 1, y^{^{'}} (0) = 2

What is (\partial)/(\partial x)(x*arccsc(x/2)) ?
The answer to (\partial)/(\partial x)(x*arccsc(x/2)) is arccsc(x/2)-2/(sqrt(x^2-4))