Issolvi v/v+1+3/v-1-6/v^2-1

$$\frac { v } { v + 1 } + \frac { 3 } { v - 1 } - \frac { 6 } { v ^ { 2 } - 1 }$$

Evalwa

$$\frac{v+3}{v+1}$$

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' v+1 u v-1 huwa \left(v-1\right)\left(v+1\right). Immultiplika \frac{v}{v+1} b'\frac{v-1}{v-1}. Immultiplika \frac{3}{v-1} b'\frac{v+1}{v+1}.

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

Billi \frac{v\left(v-1\right)}{\left(v-1\right)\left(v+1\right)} u \frac{3\left(v+1\right)}{\left(v-1\right)\left(v+1\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-multiplikazzjonijiet fi v\left(v-1\right)+3\left(v+1\right).

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

Ikkombina termini simili f'v^{2}-v+3v+3.

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

Iffattura v^{2}-1.

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

Billi \frac{v^{2}+2v+3}{\left(v-1\right)\left(v+1\right)} u \frac{6}{\left(v-1\right)\left(v+1\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

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

Ikkombina termini simili f'v^{2}+2v+3-6.

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

Iffattura l-espressjonijiet li mhumiex diġà fatturati f'\frac{v^{2}+2v-3}{\left(v-1\right)\left(v+1\right)}.

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

Annulla v-1 fin-numeratur u d-denominatur.

$$\frac{v+3}{v+1}$$

Iddifferenzja w.r.t. v

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

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' v+1 u v-1 huwa \left(v-1\right)\left(v+1\right). Immultiplika \frac{v}{v+1} b'\frac{v-1}{v-1}. Immultiplika \frac{3}{v-1} b'\frac{v+1}{v+1}.

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

Billi \frac{v\left(v-1\right)}{\left(v-1\right)\left(v+1\right)} u \frac{3\left(v+1\right)}{\left(v-1\right)\left(v+1\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-multiplikazzjonijiet fi v\left(v-1\right)+3\left(v+1\right).

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

Ikkombina termini simili f'v^{2}-v+3v+3.

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

Iffattura v^{2}-1.

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

Billi \frac{v^{2}+2v+3}{\left(v-1\right)\left(v+1\right)} u \frac{6}{\left(v-1\right)\left(v+1\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

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

Ikkombina termini simili f'v^{2}+2v+3-6.

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

Iffattura l-espressjonijiet li mhumiex diġà fatturati f'\frac{v^{2}+2v-3}{\left(v-1\right)\left(v+1\right)}.

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

Annulla v-1 fin-numeratur u d-denominatur.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v+3}{v+1})$$

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.

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

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}.

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

Agħmel l-aritmetika.

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

Espandi bl-użu ta' propjetà distributtiva.

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

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

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

Neħħi l-parenteżi mhux meħtieġa.

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

Ikkombina termini simili.

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

Naqqas 1 minn 1 u 3 minn 1.

$$\frac{-2v^{0}}{\left(v^{1}+1\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{-2v^{0}}{\left(v+1\right)^{2}}$$

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

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