Réitigh sinh(1)=

$$\sin h ( 1 ) =$$

Difreálaigh w.r.t. h

$$\cos(h)$$

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.

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

Úsáid an Fhoirmle Shuime don Síneas.

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

Fág \sin(h) as an áireamh.

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

Athscríobh an teorainn.

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

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

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

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

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

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

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

Méadaigh \cos(t)+1 faoi \cos(t)-1.

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

Baint Úsáid as Aitheantas Píotagarásach.

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

Athscríobh an teorainn.

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

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

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

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

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

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

$$\cos(h)$$

Luacháil

$$\sin(h)$$