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 xA 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
Solution to the Above Example
More References and linksRules of Differentiation of Functions in Calculus.Trigonometric Identities and Formulas. Derivatives of the Trigonometric Functions. Chain Rule of Differentiation in Calculus. |