Réitigh 2/x-2+3/x+1

$$\frac { 2 } { x - 2 } + \frac { 3 } { x + 1 }$$

Luacháil

$$\frac{5x-4}{\left(x-2\right)\left(x+1\right)}$$

Céimeanna gearra réitigh

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Is é an t-iolrach is lú coitianta de x-2 agus x+1 ná \left(x-2\right)\left(x+1\right). Méadaigh \frac{2}{x-2} faoi \frac{x+1}{x+1}. Méadaigh \frac{3}{x+1} faoi \frac{x-2}{x-2}.

$$\frac{2\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}+\frac{3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}$$

Tá an t-ainmneoir céanna ag \frac{2\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} agus \frac{3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

$$\frac{2\left(x+1\right)+3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}$$

Déan iolrúcháin in 2\left(x+1\right)+3\left(x-2\right).

$$\frac{2x+2+3x-6}{\left(x-2\right)\left(x+1\right)}$$

Cumaisc téarmaí comhchosúla in: 2x+2+3x-6.

$$\frac{5x-4}{\left(x-2\right)\left(x+1\right)}$$

Fairsingigh \left(x-2\right)\left(x+1\right)

$$\frac{5x-4}{x^{2}-x-2}$$

Difreálaigh w.r.t. x

$$\frac{-5x^{2}+8x-14}{\left(\left(x-2\right)\left(x+1\right)\right)^{2}}$$

Céimeanna ag Baint Úsáid as Riail na nDíorthach don Líon

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Is é an t-iolrach is lú coitianta de x-2 agus x+1 ná \left(x-2\right)\left(x+1\right). Méadaigh \frac{2}{x-2} faoi \frac{x+1}{x+1}. Méadaigh \frac{3}{x+1} faoi \frac{x-2}{x-2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}+\frac{3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)})$$

Tá an t-ainmneoir céanna ag \frac{2\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} agus \frac{3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(x+1\right)+3\left(x-2\right)}{\left(x-2\right)\left(x+1\right)})$$

Déan iolrúcháin in 2\left(x+1\right)+3\left(x-2\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+2+3x-6}{\left(x-2\right)\left(x+1\right)})$$

Cumaisc téarmaí comhchosúla in: 2x+2+3x-6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x-4}{\left(x-2\right)\left(x+1\right)})$$

Cuir an t-airí dáileacháin i bhfeidhm trí gach téarma de x-2 a iolrú faoi gach téarma de x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x-4}{x^{2}+x-2x-2})$$

Comhcheangail x agus -2x chun -x a fháil.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x-4}{x^{2}-x-2})$$

Do dhá fheidhm indifreáilte ar bith, is ionann díorthach líon an dá fheidhme agus an t-ainmneoir méadaithe faoi dhíorthach an uimhreora lúide an t-uimhreoir méadaithe faoi dhíorthach an ainmneora, agus iad ar fad roinnte faoin ainmneoir cearnaithe.

$$\frac{\left(x^{2}-x^{1}-2\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}-4)-\left(5x^{1}-4\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x^{1}-2)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Is ionann díorthach iltéarmaigh agus suim dhíorthaigh a théarmaí. Is ionann díorthach téarma thairisigh agus 0. Is ionann díorthach ax^{n} agus nax^{n-1}.

$$\frac{\left(x^{2}-x^{1}-2\right)\times 5x^{1-1}-\left(5x^{1}-4\right)\left(2x^{2-1}-x^{1-1}\right)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Simpligh.

$$\frac{\left(x^{2}-x^{1}-2\right)\times 5x^{0}-\left(5x^{1}-4\right)\left(2x^{1}-x^{0}\right)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Méadaigh x^{2}-x^{1}-2 faoi 5x^{0}.

$$\frac{x^{2}\times 5x^{0}-x^{1}\times 5x^{0}-2\times 5x^{0}-\left(5x^{1}-4\right)\left(2x^{1}-x^{0}\right)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Méadaigh 5x^{1}-4 faoi 2x^{1}-x^{0}.

$$\frac{x^{2}\times 5x^{0}-x^{1}\times 5x^{0}-2\times 5x^{0}-\left(5x^{1}\times 2x^{1}+5x^{1}\left(-1\right)x^{0}-4\times 2x^{1}-4\left(-1\right)x^{0}\right)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Chun cumhachtaí an bhoinn chéanna a mhéadú, suimigh a n-easpónaint.

$$\frac{5x^{2}-5x^{1}-2\times 5x^{0}-\left(5\times 2x^{1+1}+5\left(-1\right)x^{1}-4\times 2x^{1}-4\left(-1\right)x^{0}\right)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Simpligh.

$$\frac{5x^{2}-5x^{1}-10x^{0}-\left(10x^{2}-5x^{1}-8x^{1}+4x^{0}\right)}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Cuir téarmaí cosúla le chéile.

$$\frac{-5x^{2}+8x^{1}-14x^{0}}{\left(x^{2}-x^{1}-2\right)^{2}}$$

Do théarma ar bith t, t^{1}=t.

$$\frac{-5x^{2}+8x-14x^{0}}{\left(x^{2}-x-2\right)^{2}}$$

Do théarma ar bith t ach amháin 0, t^{0}=1.

$$\frac{-5x^{2}+8x-14}{\left(x^{2}-x-2\right)^{2}}$$