Proof of Derivative of cot x
The derivative of \( \cot (x)\)
is computed using the derivative of \( \sin x \) and \( \cos x \) and the quotient rule of differentiation. Examples of derivatives of cotangent composite functions are also presented along with their solutions.
Proof of the Derivative of cot x
A trigonometric identity relating \( \cot x \), \( \cos x \) and \( \sin x \) is given by
\[ \cot x = \dfrac { \cos x }{ \sin x } \]
We now use the quotient rule of differentiation way to find the derivative of \( \cot x \)
\( \displaystyle {\dfrac {d}{dx} \cot x = \dfrac {d}{dx} (\dfrac{\cos x }{\sin x}) = \dfrac{{ (\dfrac {d}{dx}\cos x) }{ \sin x }  \cos x (\dfrac {d}{dx} \sin x) }{\sin^2 x}} \)
Use the formulae for the derivative of the trigonometric functions \( \sin x \) and \( \cos x \) given by \( \dfrac {d}{dx}\cos x =  \sin x \) and \( \dfrac {d}{dx}\sin x = \cos x \) and substitute to obtain
\( \displaystyle {\dfrac {d}{dx} \cot x = \dfrac{{ \sin x \sin x }  \cos x \cos x }{\sin^2 x}} \)
Simplify
\( \displaystyle {=  \dfrac{ \sin^2 x + \cos^2 x } {\sin^2 x} =  \dfrac{ 1 }{\sin^2 x} =  \csc^2 x }\)
conclusion
\[ \displaystyle {\dfrac {d}{dx} \cot x =  \csc^2 x} \]
Graph of cot x and its Derivative
The graphs of \( \cot(x) \) and its derivative are shown below. The derivative of cot(x) is negative everywhere because cot(x) is a decreasing function.
Derivative of the Composite Function cot (u(x))
We now consider a composite function which is a function cot of another function u. Use the chain rule of differentiation to write
\( \displaystyle \dfrac{d}{dx} \cot (u(x)) = (\dfrac{d}{du} \cot u) (\dfrac{d}{dx} u ) \)
Simplify
\( =  \csc^2 u \dfrac{d}{dx} u \)
Conclusion
\[ \displaystyle \dfrac{d}{dx} \cot (u(x)) =  \csc^2 u \dfrac{d}{dx} u \]
Example
Find the derivative of the composite tan functions
 \( f(x) = \cot (x^32x+2) \)
 \( g(x) = \cot (e^x) \)
 \( h(x) = \cot (\dfrac{2}{x^3+2}) \)
Solution to the Above Example

Let \( u(x) = x^32x+2 \) and therefore \( \dfrac{d}{dx} u = \dfrac{d}{dx} (x^32x+2) = 3x^22 \) and apply the rule for the composite cot function given above.
\( \displaystyle \dfrac{d}{dx} f(x) = \csc^2 u \dfrac{d}{dx} u = \csc^2 (x^32x+2) \times (3x^22) \)
\( = (3x^22) \csc^2 (x^32x+2) \)

Let \( u(x) = e^x \) and therefore \( \dfrac{d}{dx} u = \dfrac{d}{dx} e^x = e^x \) and apply the rule of cot composite given above.
\( \displaystyle \dfrac{d}{dx} g(x) =  \csc^2 u \dfrac{d}{dx} u \)
\( =  \csc^2 (e^x) \times e^x =  e^x \csc^2 (e^x) \)

Let \( u(x) = \dfrac{2}{x^3+2} \) and therefore \( \dfrac{d}{dx} u = \dfrac{6x^2}{\left(x^3+2\right)^2} \) and apply the rule of cot composite obtained above
\( \displaystyle \dfrac{d}{dx} h(x) =  \csc^2 u \dfrac{d}{dx} u =  \csc^2 (\dfrac{2}{x^3+2}) \times \dfrac{6x^2}{\left(x^3+2\right)^2} \)
\( =  \dfrac{6x^2}{\left(x^3+2\right)^2} \csc^2 (\dfrac{2}{x^3+2}) =  \dfrac{6x^2}{\left(x^3+2\right)^2} \csc^2 (\dfrac{2}{x^3+2}) \)
More References and links
Rules of Differentiation of Functions in Calculus.
Trigonometric Identities and Formulas.
Derivatives of the Trigonometric Functions.
Chain Rule of Differentiation in Calculus.