What is the derivative of csc(x(x+cot(x))) ?

The derivative of csc(x(x+cot(x))) is -xcot(x)csc(x)-2cot^2(x)csc(x)

What is the first derivative of csc(x(x+cot(x))) ?

The first derivative of csc(x(x+cot(x))) is -xcot(x)csc(x)-2cot^2(x)csc(x)

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\frac{d}{d x} (csc (x) (x + cot (x)))

- x cot (x) csc (x) - 2 cot^{2} (x) csc (x)

Solution steps

\frac{d}{d x} (csc (x) (x + cot (x)))

Apply the Product Rule:

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

f = csc (x), g = x + cot (x)

= \frac{d}{d x} (csc (x)) (x + cot (x)) + \frac{d}{d x} (x + cot (x)) csc (x)

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

\frac{d}{d x} (x + cot (x)) = - cot^{2} (x)

= (- cot (x) csc (x)) (x + cot (x)) + (- cot^{2} (x)) csc (x)

Simplify

(- cot (x) csc (x)) (x + cot (x)) + (- cot^{2} (x)) csc (x) : - x cot (x) csc (x) - 2 cot^{2} (x) csc (x)

= - x cot (x) csc (x) - 2 cot^{2} (x) csc (x)

What is the derivative of csc(x(x+cot(x))) ?
The derivative of csc(x(x+cot(x))) is -xcot(x)csc(x)-2cot^2(x)csc(x)
What is the first derivative of csc(x(x+cot(x))) ?
The first derivative of csc(x(x+cot(x))) is -xcot(x)csc(x)-2cot^2(x)csc(x)