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Afgeleide van $$$\sin{\left(x y \right)}$$$ naar $$$y$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{dy} \left(\sin{\left(x y \right)}\right)$$$.
Oplossing
De functie $$$\sin{\left(x y \right)}$$$ is de samenstelling $$$f{\left(g{\left(y \right)} \right)}$$$ van twee functies $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ en $$$g{\left(y \right)} = x y$$$.
Pas de kettingregel $$$\frac{d}{dy} \left(f{\left(g{\left(y \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dy} \left(g{\left(y \right)}\right)$$$ toe:
$${\color{red}\left(\frac{d}{dy} \left(\sin{\left(x y \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dy} \left(x y\right)\right)}$$De afgeleide van de sinus is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dy} \left(x y\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dy} \left(x y\right)$$Keer terug naar de oorspronkelijke variabele:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dy} \left(x y\right) = \cos{\left({\color{red}\left(x y\right)} \right)} \frac{d}{dy} \left(x y\right)$$Pas de regel van de constante factor $$$\frac{d}{dy} \left(c f{\left(y \right)}\right) = c \frac{d}{dy} \left(f{\left(y \right)}\right)$$$ toe met $$$c = x$$$ en $$$f{\left(y \right)} = y$$$:
$$\cos{\left(x y \right)} {\color{red}\left(\frac{d}{dy} \left(x y\right)\right)} = \cos{\left(x y \right)} {\color{red}\left(x \frac{d}{dy} \left(y\right)\right)}$$Pas de machtsregel $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ toe met $$$n = 1$$$, met andere woorden, $$$\frac{d}{dy} \left(y\right) = 1$$$:
$$x \cos{\left(x y \right)} {\color{red}\left(\frac{d}{dy} \left(y\right)\right)} = x \cos{\left(x y \right)} {\color{red}\left(1\right)}$$Dus, $$$\frac{d}{dy} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)}$$$.
Antwoord
$$$\frac{d}{dy} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)}$$$A