Issolvi 2/x-5-5/x+3

$$\frac { 2 } { x - 5 } - \frac { 5 } { x + 3 }$$

Evalwa

$$\frac{31-3x}{\left(x-5\right)\left(x+3\right)}$$

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' x-5 u x+3 huwa \left(x-5\right)\left(x+3\right). Immultiplika \frac{2}{x-5} b'\frac{x+3}{x+3}. Immultiplika \frac{5}{x+3} b'\frac{x-5}{x-5}.

$$\frac{2\left(x+3\right)}{\left(x-5\right)\left(x+3\right)}-\frac{5\left(x-5\right)}{\left(x-5\right)\left(x+3\right)}$$

Billi \frac{2\left(x+3\right)}{\left(x-5\right)\left(x+3\right)} u \frac{5\left(x-5\right)}{\left(x-5\right)\left(x+3\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{2\left(x+3\right)-5\left(x-5\right)}{\left(x-5\right)\left(x+3\right)}$$

Agħmel il-multiplikazzjonijiet fi 2\left(x+3\right)-5\left(x-5\right).

$$\frac{2x+6-5x+25}{\left(x-5\right)\left(x+3\right)}$$

Ikkombina termini simili f'2x+6-5x+25.

$$\frac{-3x+31}{\left(x-5\right)\left(x+3\right)}$$

Espandi \left(x-5\right)\left(x+3\right).

$$\frac{-3x+31}{x^{2}-2x-15}$$

Iddifferenzja w.r.t. x

$$\frac{3x^{2}-62x+107}{x^{4}-4x^{3}-26x^{2}+60x+225}$$

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-5 u x+3 huwa \left(x-5\right)\left(x+3\right). Immultiplika \frac{2}{x-5} b'\frac{x+3}{x+3}. Immultiplika \frac{5}{x+3} b'\frac{x-5}{x-5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(x+3\right)}{\left(x-5\right)\left(x+3\right)}-\frac{5\left(x-5\right)}{\left(x-5\right)\left(x+3\right)})$$

Billi \frac{2\left(x+3\right)}{\left(x-5\right)\left(x+3\right)} u \frac{5\left(x-5\right)}{\left(x-5\right)\left(x+3\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(x+3\right)-5\left(x-5\right)}{\left(x-5\right)\left(x+3\right)})$$

Agħmel il-multiplikazzjonijiet fi 2\left(x+3\right)-5\left(x-5\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+6-5x+25}{\left(x-5\right)\left(x+3\right)})$$

Ikkombina termini simili f'2x+6-5x+25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-3x+31}{\left(x-5\right)\left(x+3\right)})$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' x-5 b'kull terminu ta' x+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-3x+31}{x^{2}+3x-5x-15})$$

Ikkombina 3x u -5x biex tikseb -2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-3x+31}{x^{2}-2x-15})$$

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.

$$\frac{\left(x^{2}-2x^{1}-15\right)\frac{\mathrm{d}}{\mathrm{d}x}(-3x^{1}+31)-\left(-3x^{1}+31\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x^{1}-15)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

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}.

$$\frac{\left(x^{2}-2x^{1}-15\right)\left(-3\right)x^{1-1}-\left(-3x^{1}+31\right)\left(2x^{2-1}-2x^{1-1}\right)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

Issimplifika.

$$\frac{\left(x^{2}-2x^{1}-15\right)\left(-3\right)x^{0}-\left(-3x^{1}+31\right)\left(2x^{1}-2x^{0}\right)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

Immultiplika x^{2}-2x^{1}-15 b'-3x^{0}.

$$\frac{x^{2}\left(-3\right)x^{0}-2x^{1}\left(-3\right)x^{0}-15\left(-3\right)x^{0}-\left(-3x^{1}+31\right)\left(2x^{1}-2x^{0}\right)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

Immultiplika -3x^{1}+31 b'2x^{1}-2x^{0}.

$$\frac{x^{2}\left(-3\right)x^{0}-2x^{1}\left(-3\right)x^{0}-15\left(-3\right)x^{0}-\left(-3x^{1}\times 2x^{1}-3x^{1}\left(-2\right)x^{0}+31\times 2x^{1}+31\left(-2\right)x^{0}\right)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

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

$$\frac{-3x^{2}-2\left(-3\right)x^{1}-15\left(-3\right)x^{0}-\left(-3\times 2x^{1+1}-3\left(-2\right)x^{1}+31\times 2x^{1}+31\left(-2\right)x^{0}\right)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

Issimplifika.

$$\frac{-3x^{2}+6x^{1}+45x^{0}-\left(-6x^{2}+6x^{1}+62x^{1}-62x^{0}\right)}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

Ikkombina termini simili.

$$\frac{3x^{2}-62x^{1}+107x^{0}}{\left(x^{2}-2x^{1}-15\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{3x^{2}-62x+107x^{0}}{\left(x^{2}-2x-15\right)^{2}}$$

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

$$\frac{3x^{2}-62x+107\times 1}{\left(x^{2}-2x-15\right)^{2}}$$

Għal kwalunkwe terminu t, t\times 1=t u 1t=t.

$$\frac{3x^{2}-62x+107}{\left(x^{2}-2x-15\right)^{2}}$$