Issolvi 5/x+1+6/x-1

$$\frac{ 5 }{ x+1 } + \frac{ 6 }{ x-1 }$$

Evalwa

$$\frac{11x+1}{x^{2}-1}$$

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' x+1 u x-1 huwa \left(x-1\right)\left(x+1\right). Immultiplika \frac{5}{x+1} b'\frac{x-1}{x-1}. Immultiplika \frac{6}{x-1} b'\frac{x+1}{x+1}.

$$\frac{5\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{6\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}$$

Billi \frac{5\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} u \frac{6\left(x+1\right)}{\left(x-1\right)\left(x+1\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

$$\frac{5\left(x-1\right)+6\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}$$

Agħmel il-multiplikazzjonijiet fi 5\left(x-1\right)+6\left(x+1\right).

$$\frac{5x-5+6x+6}{\left(x-1\right)\left(x+1\right)}$$

Ikkombina termini simili f'5x-5+6x+6.

$$\frac{11x+1}{\left(x-1\right)\left(x+1\right)}$$

Espandi \left(x-1\right)\left(x+1\right).

$$\frac{11x+1}{x^{2}-1}$$

Iddifferenzja w.r.t. x

$$-\frac{11x^{2}+2x+11}{\left(x^{2}-1\right)^{2}}$$

Passi għall-Użu ta' Regola Derivattiva għal Kwozjent

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' x+1 u x-1 huwa \left(x-1\right)\left(x+1\right). Immultiplika \frac{5}{x+1} b'\frac{x-1}{x-1}. Immultiplika \frac{6}{x-1} b'\frac{x+1}{x+1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{6\left(x+1\right)}{\left(x-1\right)\left(x+1\right)})$$

Billi \frac{5\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} u \frac{6\left(x+1\right)}{\left(x-1\right)\left(x+1\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(x-1\right)+6\left(x+1\right)}{\left(x-1\right)\left(x+1\right)})$$

Agħmel il-multiplikazzjonijiet fi 5\left(x-1\right)+6\left(x+1\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x-5+6x+6}{\left(x-1\right)\left(x+1\right)})$$

Ikkombina termini simili f'5x-5+6x+6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{11x+1}{\left(x-1\right)\left(x+1\right)})$$

Ikkunsidra li \left(x-1\right)\left(x+1\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{11x+1}{x^{2}-1^{2}})$$

Ikkalkula 1 bil-power ta' 2 u tikseb 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{11x+1}{x^{2}-1})$$

Għal kwalunkwe żewġ funzjonijiet differenzjabbli, id-derivattiv tal-kwozjent ta' żewġ funzjonijiet huwa d-denominatur immultiplikat bid-derivattiv tan-numeratur minus in-numeratur immultiplikat bid-derivattiv tad-denominatur, kollha diviżi bid-denominatur kwadrat.

$$\frac{\left(x^{2}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(11x^{1}+1)-\left(11x^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-1)}{\left(x^{2}-1\right)^{2}}$$

Id-derivattiva ta’ polynomial hija s-somma tad-derivattivi tat-termini tagħha. Id-derivattiva ta’ terminu kostanti hija 0. Id-derivattiva ta’ ax^{n} hijanax^{n-1}.

$$\frac{\left(x^{2}-1\right)\times 11x^{1-1}-\left(11x^{1}+1\right)\times 2x^{2-1}}{\left(x^{2}-1\right)^{2}}$$

Agħmel l-aritmetika.

$$\frac{\left(x^{2}-1\right)\times 11x^{0}-\left(11x^{1}+1\right)\times 2x^{1}}{\left(x^{2}-1\right)^{2}}$$

Espandi bl-użu ta' propjetà distributtiva.

$$\frac{x^{2}\times 11x^{0}-11x^{0}-\left(11x^{1}\times 2x^{1}+2x^{1}\right)}{\left(x^{2}-1\right)^{2}}$$

Biex timmultiplika l-qawwa tal-istess bażi, żid l-esponenti tagħhom.

$$\frac{11x^{2}-11x^{0}-\left(11\times 2x^{1+1}+2x^{1}\right)}{\left(x^{2}-1\right)^{2}}$$

Agħmel l-aritmetika.

$$\frac{11x^{2}-11x^{0}-\left(22x^{2}+2x^{1}\right)}{\left(x^{2}-1\right)^{2}}$$

Neħħi l-parenteżi mhux meħtieġa.

$$\frac{11x^{2}-11x^{0}-22x^{2}-2x^{1}}{\left(x^{2}-1\right)^{2}}$$

Ikkombina termini simili.

$$\frac{\left(11-22\right)x^{2}-11x^{0}-2x^{1}}{\left(x^{2}-1\right)^{2}}$$

Naqqas 22 minn 11.

$$\frac{-11x^{2}-11x^{0}-2x^{1}}{\left(x^{2}-1\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{-11x^{2}-11x^{0}-2x}{\left(x^{2}-1\right)^{2}}$$

Għal kwalunkwe terminu t ħlief 0, t^{0}=1.

$$\frac{-11x^{2}-11-2x}{\left(x^{2}-1\right)^{2}}$$