Réitigh f(x)=siny

$$f ( x ) = \sin y$$

Difreálaigh w.r.t. y

$$\cos(y)$$

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.

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

Úsáid an Fhoirmle Shuime don Síneas.

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

Fág \sin(y) as an áireamh.

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

Athscríobh an teorainn.

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

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

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

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

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

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

$$\cos(y)$$

Luacháil

$$\sin(y)$$