Issolvi 7/x+3+2/x+2

$$\frac{ 7 }{ x+3 } + \frac{ 2 }{ x+2 }$$

Evalwa

$$\frac{9x+20}{\left(x+2\right)\left(x+3\right)}$$

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' x+3 u x+2 huwa \left(x+2\right)\left(x+3\right). Immultiplika \frac{7}{x+3} b'\frac{x+2}{x+2}. Immultiplika \frac{2}{x+2} b'\frac{x+3}{x+3}.

$$\frac{7\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}+\frac{2\left(x+3\right)}{\left(x+2\right)\left(x+3\right)}$$

Billi \frac{7\left(x+2\right)}{\left(x+2\right)\left(x+3\right)} u \frac{2\left(x+3\right)}{\left(x+2\right)\left(x+3\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

$$\frac{7\left(x+2\right)+2\left(x+3\right)}{\left(x+2\right)\left(x+3\right)}$$

Agħmel il-multiplikazzjonijiet fi 7\left(x+2\right)+2\left(x+3\right).

$$\frac{7x+14+2x+6}{\left(x+2\right)\left(x+3\right)}$$

Ikkombina termini simili f'7x+14+2x+6.

$$\frac{9x+20}{\left(x+2\right)\left(x+3\right)}$$

Espandi \left(x+2\right)\left(x+3\right).

$$\frac{9x+20}{x^{2}+5x+6}$$

Iddifferenzja w.r.t. x

$$-\frac{9x^{2}+40x+46}{\left(\left(x+2\right)\left(x+3\right)\right)^{2}}$$

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+3 u x+2 huwa \left(x+2\right)\left(x+3\right). Immultiplika \frac{7}{x+3} b'\frac{x+2}{x+2}. Immultiplika \frac{2}{x+2} b'\frac{x+3}{x+3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}+\frac{2\left(x+3\right)}{\left(x+2\right)\left(x+3\right)})$$

Billi \frac{7\left(x+2\right)}{\left(x+2\right)\left(x+3\right)} u \frac{2\left(x+3\right)}{\left(x+2\right)\left(x+3\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7\left(x+2\right)+2\left(x+3\right)}{\left(x+2\right)\left(x+3\right)})$$

Agħmel il-multiplikazzjonijiet fi 7\left(x+2\right)+2\left(x+3\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7x+14+2x+6}{\left(x+2\right)\left(x+3\right)})$$

Ikkombina termini simili f'7x+14+2x+6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x+20}{\left(x+2\right)\left(x+3\right)})$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' x+2 b'kull terminu ta' x+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x+20}{x^{2}+3x+2x+6})$$

Ikkombina 3x u 2x biex tikseb 5x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x+20}{x^{2}+5x+6})$$

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.

$$\frac{\left(x^{2}+5x^{1}+6\right)\frac{\mathrm{d}}{\mathrm{d}x}(9x^{1}+20)-\left(9x^{1}+20\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+5x^{1}+6)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

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}.

$$\frac{\left(x^{2}+5x^{1}+6\right)\times 9x^{1-1}-\left(9x^{1}+20\right)\left(2x^{2-1}+5x^{1-1}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Issimplifika.

$$\frac{\left(x^{2}+5x^{1}+6\right)\times 9x^{0}-\left(9x^{1}+20\right)\left(2x^{1}+5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Immultiplika x^{2}+5x^{1}+6 b'9x^{0}.

$$\frac{x^{2}\times 9x^{0}+5x^{1}\times 9x^{0}+6\times 9x^{0}-\left(9x^{1}+20\right)\left(2x^{1}+5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Immultiplika 9x^{1}+20 b'2x^{1}+5x^{0}.

$$\frac{x^{2}\times 9x^{0}+5x^{1}\times 9x^{0}+6\times 9x^{0}-\left(9x^{1}\times 2x^{1}+9x^{1}\times 5x^{0}+20\times 2x^{1}+20\times 5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

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

$$\frac{9x^{2}+5\times 9x^{1}+6\times 9x^{0}-\left(9\times 2x^{1+1}+9\times 5x^{1}+20\times 2x^{1}+20\times 5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Issimplifika.

$$\frac{9x^{2}+45x^{1}+54x^{0}-\left(18x^{2}+45x^{1}+40x^{1}+100x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Ikkombina termini simili.

$$\frac{-9x^{2}-40x^{1}-46x^{0}}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{-9x^{2}-40x-46x^{0}}{\left(x^{2}+5x+6\right)^{2}}$$

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

$$\frac{-9x^{2}-40x-46}{\left(x^{2}+5x+6\right)^{2}}$$