Issolvi d/dθleft(cos(θ)right)

$$\frac{d}{d \theta } \left( \cos ( \theta ) \right)$$

Evalwa

$$-\sin(\theta )$$

Passi għall-Użu ta' Definizzjoni ta' Dervattiv

Għall-funzjoni f\left(x\right), id-derivattiv huwa l-limitu ta' \frac{f\left(x+h\right)-f\left(x\right)}{h} billi h jmur għal 0, jekk jeżisti dak il-limitu.

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

Uża l-Formula tas-Somma għal Cosine.

$$\lim_{h\to 0}\frac{\cos(h+\theta )-\cos(\theta )}{h}$$

Iffattura 'l barra \cos(\theta ).

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

Erġa' ikteb il-limitu.

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

Uża l-fatt li \theta huwa kostanti meta taħdem il-limiti bħala h jmorru għal 0.

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

Il-limitu \lim_{\theta \to 0}\frac{\sin(\theta )}{\theta } huwa 1.

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

Biex tevalwa l-limitu \lim_{h\to 0}\frac{\cos(h)-1}{h}, l-ewwel immultiplika n-numeratur u d-denominatur b'\cos(h)+1.

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

Immultiplika \cos(h)+1 b'\cos(h)-1.

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

Uża l-Pythagorean Identity.

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

Erġa' ikteb il-limitu.

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

Il-limitu \lim_{\theta \to 0}\frac{\sin(\theta )}{\theta } huwa 1.

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

Uża l-fatt li \frac{\sin(h)}{\cos(h)+1} huwa kontinwu f'0.

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

Issostitwixxi l-valur 0 fl-espressjoni \cos(\theta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\sin(\theta ).

$$-\sin(\theta )$$

Iddifferenzja w.r.t. θ

$$-\cos(\theta )$$