Proof of the Derivative of csc x

Graph of csc x and its Derivative

The graphs of \( \csc(x) \) and its derivative are shown below.

Derivative of the Composite Function csc (u(x))

Example 1
Find the derivative of the composite csc functions

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

Solution to Example 1

1. 
   Let \( u(x) = -x^3+3 \) and therefore \( \dfrac{d}{dx} u = \dfrac{d}{dx} (-x^3+3) = -3x^2 \) and apply the rule for the composite csc function given above
   
   \( \displaystyle \dfrac{d}{dx} f(x) = - \cot u \csc u \dfrac{d}{dx} u = - \cot (-x^3+3) \csc (-x^3+3) \times (-3x^2) \)
   
   \( = 3x^2 \; \cot (-x^3+3) \; \csc (-x^3+3) \)
   
   
2. 
   Let \( u(x) = \cos x \) and therefore \( \dfrac{d}{dx} u = \dfrac{d}{dx} \cos x = - \sin x \) and apply the above rule of differentiation for the composite csc function
   
   \( \displaystyle \dfrac{d}{dx} g(x) = - \cot u \csc u \dfrac{d}{dx} u = - \cot (\cos x) \csc (\cos x) \times (- \sin x) \)
   
   \( = \sin x \;\cot (\cos x) \; \csc (\cos x) \)
   
   
3. 
   Let \( u(x) = \dfrac{1}{x^2+1} \) and therefore \( \dfrac{d}{dx} u = -\dfrac{2x}{(x^2+1)^2} \) and apply the rule of differentiation for the composite csc function obtained above
   
   \( \displaystyle \dfrac{d}{dx} h(x) = - \cot u \csc u \dfrac{d}{dx} u = - \cot (\dfrac{1}{x^2+1}) \csc (\dfrac{1}{x^2+1}) \times (-\dfrac{2x}{(x^2+1)^2}) \)
   
   \( = \dfrac{2x}{(x^2+1)^2} \; \cot (\dfrac{1}{x^2+1}) \;\csc (\dfrac{1}{x^2+1}) \)
   
   

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