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Derivatan av $$$\csc^{2}{\left(x \right)}$$$
Relaterade kalkylatorer: Kalkylator för logaritmisk derivering, Räknare för implicit derivering med steg
Din inmatning
Bestäm $$$\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right)$$$.
Lösning
Funktionen $$$\csc^{2}{\left(x \right)}$$$ är sammansättningen $$$f{\left(g{\left(x \right)} \right)}$$$ av två funktioner $$$f{\left(u \right)} = u^{2}$$$ och $$$g{\left(x \right)} = \csc{\left(x \right)}$$$.
Tillämpa kedjeregeln $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)}$$Tillämpa potensregeln $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ med $$$n = 2$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right)$$Återgå till den ursprungliga variabeln:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right) = 2 {\color{red}\left(\csc{\left(x \right)}\right)} \frac{d}{dx} \left(\csc{\left(x \right)}\right)$$Derivatan av kosekansen är $$$\frac{d}{dx} \left(\csc{\left(x \right)}\right) = - \cot{\left(x \right)} \csc{\left(x \right)}$$$:
$$2 \csc{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\csc{\left(x \right)}\right)\right)} = 2 \csc{\left(x \right)} {\color{red}\left(- \cot{\left(x \right)} \csc{\left(x \right)}\right)}$$Alltså, $$$\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right) = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$$.
Svar
$$$\frac{d}{dx} \left(\csc^{2}{\left(x \right)}\right) = - 2 \cot{\left(x \right)} \csc^{2}{\left(x \right)}$$$A