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Finding the Derivative of cot(x)

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson we will find the derivative of cotangent by using trigonometric identities and the quotient rule.

Plan of Attack

The derivative of cotangent isn't one of those pieces of information people commonly keep in their heads. The derivatives of sine and cosine, however, are much more commonly known (see table). To solve this problem we'll (1) use trigonometric identities to express cotangent as a ratio of sine and cosine, and (2) use the quotient rule to find the derivative of that ratio.

Derivative of sin(x) cos(x)
Derivative of cos(x) -sin(x)

Solution

1. Expressing cotangent as a ratio of sine and cosine.

In case you don't remember, here are some common trigonometric identities.

tan(x) = sin(x) / cos(x)
csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x) = cos(x) / sin(x)

Instead of solving for the derivative of cotangent, we'll instead take the derivative of cosine over sine.


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2. Using the quotient rule to find the derivative of the cosine over sine.

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