Rešite d/dxleft(sin(x)right)

$$\frac{d}{d x } \left( \sin ( x ) \right)$$

Ovrednoti

$$\cos(x)$$

Koraki z uporabo definicije odvoda

Za funkcijo f\left(x\right) je odvod limita funkcije \frac{f\left(x+h\right)-f\left(x\right)}{h}, saj gre h v 0, če ta limita obstaja.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\sin(x))=\left(\lim_{h\to 0}\frac{\sin(x+h)-\sin(x)}{h}\right)$$

Uporabite formulo za sinus vsote.

$$\lim_{h\to 0}\frac{\sin(x+h)-\sin(x)}{h}$$

Faktorizirajte \sin(x).

$$\lim_{h\to 0}\frac{\sin(x)\left(\cos(h)-1\right)+\cos(x)\sin(h)}{h}$$

Znova napišite limito.

$$\left(\lim_{h\to 0}\sin(x)\right)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\left(\lim_{h\to 0}\cos(x)\right)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$

Uporabite dejstvo, da je x konstanta, kadar računate limite, saj gre h v 0.

$$\sin(x)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$

Limita \lim_{x\to 0}\frac{\sin(x)}{x} je 1.

$$\sin(x)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x)$$

Če želite ovrednotiti limite \lim_{h\to 0}\frac{\cos(h)-1}{h}, najprej pomnožite števec in imenovalec s \cos(h)+1.

$$\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)=\left(\lim_{h\to 0}\frac{\left(\cos(h)-1\right)\left(\cos(h)+1\right)}{h\left(\cos(h)+1\right)}\right)$$

Pomnožite \cos(h)+1 s/z \cos(h)-1.

$$\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}$$

Uporabite Pitagorovo identiteto.

$$\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}$$

Znova napišite limito.

$$\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$

Limita \lim_{x\to 0}\frac{\sin(x)}{x} je 1.

$$-\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$

Uporabite dejstvo, da je funkcija \frac{\sin(h)}{\cos(h)+1} zvezna pri 0.

$$\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)=0$$

Vstavite vrednost 0 v izraz \sin(x)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x).

$$\cos(x)$$

Odvajajte w.r.t. x

$$-\sin(x)$$