Scipeáil chuig an bpríomhábhar
Difreálaigh w.r.t. x
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Luacháil
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\frac{\mathrm{d}}{\mathrm{d}x}(\sin(x))=\left(\lim_{h\to 0}\frac{\sin(x+h)-\sin(x)}{h}\right)
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.
\lim_{h\to 0}\frac{\sin(x+h)-\sin(x)}{h}
Úsáid an Fhoirmle Shuime don Síneas.
\lim_{h\to 0}\frac{\sin(x)\left(\cos(h)-1\right)+\cos(x)\sin(h)}{h}
Fág \sin(x) as an áireamh.
\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)
Athscríobh an teorainn.
\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)
Bain leas as an rud é go bhfuil x ina thairiseach agus teorainneacha á ríomh agus h ag dul go 0.
\sin(x)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x)
Is ionann teorainn \lim_{x\to 0}\frac{\sin(x)}{x} agus 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)
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.
\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}
Méadaigh \cos(h)+1 faoi \cos(h)-1.
\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}
Baint Úsáid as Aitheantas Píotagarásach.
\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)
Athscríobh an teorainn.
-\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)
Is ionann teorainn \lim_{x\to 0}\frac{\sin(x)}{x} agus 1.
\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)=0
Bain leas as an rud go bhfuil \frac{\sin(h)}{\cos(h)+1} leanúnach ag 0.
\cos(x)
Ionadaigh an luach 0 isteach sa slonn \sin(x)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(x).