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
\[ \displaystyle \dfrac{d}{dx} \cot (u(x)) = - \csc^2 u \dfrac{d}{dx} u \]
Example
Find the derivative of the composite tan functions
\( f(x) = \cot (x^3-2x+2) \)
\( g(x) = \cot (e^x) \)
\( h(x) = \cot (\dfrac{-2}{x^3+2}) \)
Solution to the Above Example
Let \( u(x) = x^3-2x+2 \) and therefore \( \dfrac{d}{dx} u = \dfrac{d}{dx} (x^3-2x+2) = 3x^2-2 \) and apply the rule for the composite cot function given above.
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