Réitigh 4x/x+4+1/x-4

$$\frac { 4 x } { x + 4 } + \frac { 1 } { x - 4 }$$

Luacháil

$$\frac{4x^{2}-15x+4}{x^{2}-16}$$

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+4 agus x-4 ná \left(x-4\right)\left(x+4\right). Méadaigh \frac{4x}{x+4} faoi \frac{x-4}{x-4}. Méadaigh \frac{1}{x-4} faoi \frac{x+4}{x+4}.

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

Tá an t-ainmneoir céanna ag \frac{4x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)} agus \frac{x+4}{\left(x-4\right)\left(x+4\right)} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

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

Déan iolrúcháin in 4x\left(x-4\right)+x+4.

$$\frac{4x^{2}-16x+x+4}{\left(x-4\right)\left(x+4\right)}$$

Cumaisc téarmaí comhchosúla in: 4x^{2}-16x+x+4.

$$\frac{4x^{2}-15x+4}{\left(x-4\right)\left(x+4\right)}$$

Fairsingigh \left(x-4\right)\left(x+4\right)

$$\frac{4x^{2}-15x+4}{x^{2}-16}$$

Difreálaigh w.r.t. x

$$\frac{\left(3x-20\right)\left(5x-12\right)}{\left(x^{2}-16\right)^{2}}$$

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+4 agus x-4 ná \left(x-4\right)\left(x+4\right). Méadaigh \frac{4x}{x+4} faoi \frac{x-4}{x-4}. Méadaigh \frac{1}{x-4} faoi \frac{x+4}{x+4}.

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

Tá an t-ainmneoir céanna ag \frac{4x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)} agus \frac{x+4}{\left(x-4\right)\left(x+4\right)} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

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

Déan iolrúcháin in 4x\left(x-4\right)+x+4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}-16x+x+4}{\left(x-4\right)\left(x+4\right)})$$

Cumaisc téarmaí comhchosúla in: 4x^{2}-16x+x+4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}-15x+4}{\left(x-4\right)\left(x+4\right)})$$

Mar shampla \left(x-4\right)\left(x+4\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}-15x+4}{x^{2}-4^{2}})$$

Ríomh cumhacht 4 de 2 agus faigh 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}-15x+4}{x^{2}-16})$$

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.

$$\frac{\left(x^{2}-16\right)\frac{\mathrm{d}}{\mathrm{d}x}(4x^{2}-15x^{1}+4)-\left(4x^{2}-15x^{1}+4\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-16)}{\left(x^{2}-16\right)^{2}}$$

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}.

$$\frac{\left(x^{2}-16\right)\left(2\times 4x^{2-1}-15x^{1-1}\right)-\left(4x^{2}-15x^{1}+4\right)\times 2x^{2-1}}{\left(x^{2}-16\right)^{2}}$$

Simpligh.

$$\frac{\left(x^{2}-16\right)\left(8x^{1}-15x^{0}\right)-\left(4x^{2}-15x^{1}+4\right)\times 2x^{1}}{\left(x^{2}-16\right)^{2}}$$

Méadaigh x^{2}-16 faoi 8x^{1}-15x^{0}.

$$\frac{x^{2}\times 8x^{1}+x^{2}\left(-15\right)x^{0}-16\times 8x^{1}-16\left(-15\right)x^{0}-\left(4x^{2}-15x^{1}+4\right)\times 2x^{1}}{\left(x^{2}-16\right)^{2}}$$

Méadaigh 4x^{2}-15x^{1}+4 faoi 2x^{1}.

$$\frac{x^{2}\times 8x^{1}+x^{2}\left(-15\right)x^{0}-16\times 8x^{1}-16\left(-15\right)x^{0}-\left(4x^{2}\times 2x^{1}-15x^{1}\times 2x^{1}+4\times 2x^{1}\right)}{\left(x^{2}-16\right)^{2}}$$

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

$$\frac{8x^{2+1}-15x^{2}-16\times 8x^{1}-16\left(-15\right)x^{0}-\left(4\times 2x^{2+1}-15\times 2x^{1+1}+4\times 2x^{1}\right)}{\left(x^{2}-16\right)^{2}}$$

Simpligh.

$$\frac{8x^{3}-15x^{2}-128x^{1}+240x^{0}-\left(8x^{3}-30x^{2}+8x^{1}\right)}{\left(x^{2}-16\right)^{2}}$$

Cuir téarmaí cosúla le chéile.

$$\frac{15x^{2}-136x^{1}+240x^{0}}{\left(x^{2}-16\right)^{2}}$$

Do théarma ar bith t, t^{1}=t.

$$\frac{15x^{2}-136x+240x^{0}}{\left(x^{2}-16\right)^{2}}$$

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

$$\frac{15x^{2}-136x+240\times 1}{\left(x^{2}-16\right)^{2}}$$

Do théarma ar bith t, t\times 1=t agus 1t=t.

$$\frac{15x^{2}-136x+240}{\left(x^{2}-16\right)^{2}}$$