Réitigh d/dθleft(cos(θ)right)

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

Luacháil

$$-\sin(\theta )$$

Céimeanna ag Baint Úsáid as Sainiú Díorthaigh

Do fheidhm f\left(x\right), is ionann an díorthach agus teorainn \frac{f\left(x+h\right)-f\left(x\right)}{h} toisc go dtéann h go 0, más ann don teorainn sin.

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

Úsáid an Fhoirmle Shuime don Chomhshíneas.

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

Fág \cos(\theta ) as an áireamh.

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

Athscríobh an teorainn.

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

Bain leas as an rud é go bhfuil \theta ina thairiseach agus teorainneacha á ríomh agus h ag dul go 0.

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

Is ionann teorainn \lim_{\theta \to 0}\frac{\sin(\theta )}{\theta } agus 1.

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

Chun an teorainn \lim_{h\to 0}\frac{\cos(h)-1}{h} a luacháil, méadaigh an t-uimhreoir agus an t-ainmneoir faoi \cos(h)+1 ar dtús.

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

Méadaigh \cos(h)+1 faoi \cos(h)-1.

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

Baint Úsáid as Aitheantas Píotagarásach.

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

Athscríobh an teorainn.

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

Is ionann teorainn \lim_{\theta \to 0}\frac{\sin(\theta )}{\theta } agus 1.

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

Bain leas as an rud go bhfuil \frac{\sin(h)}{\cos(h)+1} leanúnach ag 0.

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

Ionadaigh an luach 0 isteach sa slonn \cos(\theta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)-\sin(\theta ).

$$-\sin(\theta )$$

Difreálaigh w.r.t. θ

$$-\cos(\theta )$$