How to Find the Derivative of cot(x) and Express it in Terms of sine and cosine

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To find the derivative of cot(x), we can rewrite it in terms of sine and cosine functions

To find the derivative of cot(x), we can rewrite it in terms of sine and cosine functions.

cot(x) = cos(x) / sin(x)

Now, we can differentiate the function using the quotient rule.

Let f(x) = cos(x) and g(x) = sin(x).

Using the quotient rule: (f'(x)g(x) – g'(x)f(x)) / (g(x))^2

Taking the derivatives of f(x) and g(x):

f'(x) = -sin(x)
g'(x) = cos(x)

Plugging these values into the quotient rule:

((-sin(x))(sin(x)) – (cos(x))(cos(x))) / (sin(x))^2

Simplifying the numerator:

(-sin^2(x) – cos^2(x)) / sin^2(x)

Using the trigonometric identity: sin^2(x) + cos^2(x) = 1

The numerator simplifies to:

-1

So, the derivative of cot(x) is:

-1 / sin^2(x)

Alternatively, we can also express the derivative as:

-csc^2(x) or -cosec^2(x)

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