|
|
|
|
|
|
|
|
|
|
|
|
Derivatan av $$$\sin{\left(5 \theta \right)}$$$
Relaterade kalkylatorer: Kalkylator för logaritmisk derivering, Räknare för implicit derivering med steg
Din inmatning
Bestäm $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)$$$.
Lösning
Funktionen $$$\sin{\left(5 \theta \right)}$$$ är sammansättningen $$$f{\left(g{\left(\theta \right)} \right)}$$$ av två funktioner $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ och $$$g{\left(\theta \right)} = 5 \theta$$$.
Tillämpa kedjeregeln $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$:
$${\color{red}\left(\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{d\theta} \left(5 \theta\right)\right)}$$Derivatan av sinus är $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{d\theta} \left(5 \theta\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{d\theta} \left(5 \theta\right)$$Återgå till den ursprungliga variabeln:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right) = \cos{\left({\color{red}\left(5 \theta\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right)$$Tillämpa konstantfaktorregeln $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ med $$$c = 5$$$ och $$$f{\left(\theta \right)} = \theta$$$:
$$\cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(5 \theta\right)\right)} = \cos{\left(5 \theta \right)} {\color{red}\left(5 \frac{d}{d\theta} \left(\theta\right)\right)}$$Tillämpa potensregeln $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$ med $$$n = 1$$$, det vill säga $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:
$$5 \cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)} = 5 \cos{\left(5 \theta \right)} {\color{red}\left(1\right)}$$Alltså, $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$.
Svar
$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$A