Step by step solution to "How do you calculate cotx derivative?"

Marques Flynn

Answered question

2022-11-24

How do you calculate

\cot (x)

derivative?

Henry Arellano

Beginner2022-11-25Added 12 answers

Step 1: Put the quotient rule of differentiation to use.
We know that

\cot (x) = \frac{\cos (x)}{\sin (x)}

the differentiation quotient rule being applied

\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}

putting

u = \cos (x)

and

v = \sin (x)

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

Step 2: (Solve for differentiation)

\frac{d}{d x} (\frac{\cos (x)}{\sin (x)}) = \frac{\sin (x) \times - \sin (x) - \cos (x) \times \cos (x)}{\sin^{2} (x)} (∴ \frac{d}{d x} \cos (x) = - \sin x a n d \frac{d}{d x} \sin (x) = \cos (x)) \frac{d}{d x} (\frac{\cos (x)}{\sin (x)}) = \frac{- \sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x)} \frac{d}{d x} (\frac{\cos (x)}{\sin (x)}) = \frac{- (\sin^{2} (x) + \cos^{2} (x))}{\sin^{2} (x)} \frac{d}{d x} (c o t (x)) = \frac{- 1}{\sin^{2} (x)} (∴ \sin^{2} (x) + \cos^{2} (x) = 1) \frac{d}{d x} (c o t (x)) = - c o s e c^{2} (x) (∴ \frac{1}{\sin (x)} = \csc (x))

Hence, the derivative of

\cot (x)

- c o s e c^{2} (x)

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