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

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))

Example
Find the derivative of the composite tan functions

1. \( f(x) = \cot (x^3-2x+2) \)
2. \( g(x) = \cot (e^x) \)
3. \( h(x) = \cot (\dfrac{-2}{x^3+2}) \)

Solution to the Above Example

1. 
   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.
   
   \( \displaystyle \dfrac{d}{dx} f(x) = -\csc^2 u \dfrac{d}{dx} u = -\csc^2 (x^3-2x+2) \times (3x^2-2) \)
   \( = -(3x^2-2) \csc^2 (x^3-2x+2) \)
   
   
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.
   
   \( \displaystyle \dfrac{d}{dx} g(x) = - \csc^2 u \dfrac{d}{dx} u \)
   \( = - \csc^2 (e^x) \times e^x = - e^x \csc^2 (e^x) \)
   
   
3. 
   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
   
   \( \displaystyle \dfrac{d}{dx} h(x) = - \csc^2 u \dfrac{d}{dx} u = - \csc^2 (\dfrac{-2}{x^3+2}) \times \dfrac{6x^2}{\left(x^3+2\right)^2} \)
   \( = - \dfrac{6x^2}{\left(x^3+2\right)^2} \csc^2 (\dfrac{-2}{x^3+2}) = - \dfrac{6x^2}{\left(x^3+2\right)^2} \csc^2 (\dfrac{2}{x^3+2}) \)
   
   

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