|
|
|
|
|
|
|
|
|
|
|
|
Derivative of $$$\cot{\left(\frac{x}{2} \right)}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dx} \left(\cot{\left(\frac{x}{2} \right)}\right)$$$.
Solution
The function $$$\cot{\left(\frac{x}{2} \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \cot{\left(u \right)}$$$ and $$$g{\left(x \right)} = \frac{x}{2}$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cot{\left(\frac{x}{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cot{\left(u \right)}\right) \frac{d}{dx} \left(\frac{x}{2}\right)\right)}$$The derivative of the cotangent is $$$\frac{d}{du} \left(\cot{\left(u \right)}\right) = - \csc^{2}{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\cot{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{x}{2}\right) = {\color{red}\left(- \csc^{2}{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{x}{2}\right)$$Return to the old variable:
$$- \csc^{2}{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{x}{2}\right) = - \csc^{2}{\left({\color{red}\left(\frac{x}{2}\right)} \right)} \frac{d}{dx} \left(\frac{x}{2}\right)$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = \frac{1}{2}$$$ and $$$f{\left(x \right)} = x$$$:
$$- \csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{x}{2}\right)\right)} = - \csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(\frac{\frac{d}{dx} \left(x\right)}{2}\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- \frac{\csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{2} = - \frac{\csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(1\right)}}{2}$$Simplify:
$$- \frac{\csc^{2}{\left(\frac{x}{2} \right)}}{2} = \frac{1}{\cos{\left(x \right)} - 1}$$Thus, $$$\frac{d}{dx} \left(\cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\cos{\left(x \right)} - 1}$$$.
Answer
$$$\frac{d}{dx} \left(\cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\cos{\left(x \right)} - 1}$$$A