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

The derivative of (cos(x)/(sin^2(x))) is csc(x)-2csc^3(x)

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

The first derivative of (cos(x)/(sin^2(x))) is csc(x)-2csc^3(x)

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

csc (x) - 2 csc^{3} (x)

Solution steps

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

Simplify

\frac{cos ( x )}{sin ^{2} ( x )} : cot (x) csc (x)

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

(- csc^{2} (x)) csc (x) + (- cot (x) csc (x)) cot (x) : csc (x) - 2 csc^{3} (x)

= csc (x) - 2 csc^{3} (x)

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