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