Issolvi v/v^2+17v+72-8/v^2+15v+56

$$\frac { v } { v ^ { 2 } + 17 v + 72 } - \frac { 8 } { v ^ { 2 } + 15 v + 56 }$$

Evalwa

$$\frac{v-9}{\left(v+7\right)\left(v+9\right)}$$

Passi Qosra tas-Soluzzjonijiet

Iffattura v^{2}+17v+72. Iffattura v^{2}+15v+56.

$$\frac{v}{\left(v+8\right)\left(v+9\right)}-\frac{8}{\left(v+7\right)\left(v+8\right)}$$

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' \left(v+8\right)\left(v+9\right) u \left(v+7\right)\left(v+8\right) huwa \left(v+7\right)\left(v+8\right)\left(v+9\right). Immultiplika \frac{v}{\left(v+8\right)\left(v+9\right)} b'\frac{v+7}{v+7}. Immultiplika \frac{8}{\left(v+7\right)\left(v+8\right)} b'\frac{v+9}{v+9}.

$$\frac{v\left(v+7\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}-\frac{8\left(v+9\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}$$

Billi \frac{v\left(v+7\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)} u \frac{8\left(v+9\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{v\left(v+7\right)-8\left(v+9\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}$$

Agħmel il-multiplikazzjonijiet fi v\left(v+7\right)-8\left(v+9\right).

$$\frac{v^{2}+7v-8v-72}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}$$

Ikkombina termini simili f'v^{2}+7v-8v-72.

$$\frac{v^{2}-v-72}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}$$

Iffattura l-espressjonijiet li mhumiex diġà fatturati f'\frac{v^{2}-v-72}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}.

$$\frac{\left(v-9\right)\left(v+8\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}$$

Annulla v+8 fin-numeratur u d-denominatur.

$$\frac{v-9}{\left(v+7\right)\left(v+9\right)}$$

Espandi \left(v+7\right)\left(v+9\right).

$$\frac{v-9}{v^{2}+16v+63}$$

Iddifferenzja w.r.t. v

$$\frac{207+18v-v^{2}}{v^{4}+32v^{3}+382v^{2}+2016v+3969}$$

Passi għall-Użu ta' Regola Derivattiva għal Kwozjent

Iffattura v^{2}+17v+72. Iffattura v^{2}+15v+56.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v}{\left(v+8\right)\left(v+9\right)}-\frac{8}{\left(v+7\right)\left(v+8\right)})$$

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' \left(v+8\right)\left(v+9\right) u \left(v+7\right)\left(v+8\right) huwa \left(v+7\right)\left(v+8\right)\left(v+9\right). Immultiplika \frac{v}{\left(v+8\right)\left(v+9\right)} b'\frac{v+7}{v+7}. Immultiplika \frac{8}{\left(v+7\right)\left(v+8\right)} b'\frac{v+9}{v+9}.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v\left(v+7\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}-\frac{8\left(v+9\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)})$$

Billi \frac{v\left(v+7\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)} u \frac{8\left(v+9\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v\left(v+7\right)-8\left(v+9\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)})$$

Agħmel il-multiplikazzjonijiet fi v\left(v+7\right)-8\left(v+9\right).

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v^{2}+7v-8v-72}{\left(v+7\right)\left(v+8\right)\left(v+9\right)})$$

Ikkombina termini simili f'v^{2}+7v-8v-72.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v^{2}-v-72}{\left(v+7\right)\left(v+8\right)\left(v+9\right)})$$

Iffattura l-espressjonijiet li mhumiex diġà fatturati f'\frac{v^{2}-v-72}{\left(v+7\right)\left(v+8\right)\left(v+9\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{\left(v-9\right)\left(v+8\right)}{\left(v+7\right)\left(v+8\right)\left(v+9\right)})$$

Annulla v+8 fin-numeratur u d-denominatur.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v-9}{\left(v+7\right)\left(v+9\right)})$$

Uża l-propjetà distributtiva biex timmultiplika v+7 b'v+9 u kkombina termini simili.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v-9}{v^{2}+16v+63})$$

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.

$$\frac{\left(v^{2}+16v^{1}+63\right)\frac{\mathrm{d}}{\mathrm{d}v}(v^{1}-9)-\left(v^{1}-9\right)\frac{\mathrm{d}}{\mathrm{d}v}(v^{2}+16v^{1}+63)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

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}.

$$\frac{\left(v^{2}+16v^{1}+63\right)v^{1-1}-\left(v^{1}-9\right)\left(2v^{2-1}+16v^{1-1}\right)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

Issimplifika.

$$\frac{\left(v^{2}+16v^{1}+63\right)v^{0}-\left(v^{1}-9\right)\left(2v^{1}+16v^{0}\right)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

Immultiplika v^{2}+16v^{1}+63 b'v^{0}.

$$\frac{v^{2}v^{0}+16v^{1}v^{0}+63v^{0}-\left(v^{1}-9\right)\left(2v^{1}+16v^{0}\right)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

Immultiplika v^{1}-9 b'2v^{1}+16v^{0}.

$$\frac{v^{2}v^{0}+16v^{1}v^{0}+63v^{0}-\left(v^{1}\times 2v^{1}+v^{1}\times 16v^{0}-9\times 2v^{1}-9\times 16v^{0}\right)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

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

$$\frac{v^{2}+16v^{1}+63v^{0}-\left(2v^{1+1}+16v^{1}-9\times 2v^{1}-9\times 16v^{0}\right)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

Issimplifika.

$$\frac{v^{2}+16v^{1}+63v^{0}-\left(2v^{2}+16v^{1}-18v^{1}-144v^{0}\right)}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

Ikkombina termini simili.

$$\frac{-v^{2}+18v^{1}+207v^{0}}{\left(v^{2}+16v^{1}+63\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{-v^{2}+18v+207v^{0}}{\left(v^{2}+16v+63\right)^{2}}$$

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

$$\frac{-v^{2}+18v+207\times 1}{\left(v^{2}+16v+63\right)^{2}}$$

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

$$\frac{-v^{2}+18v+207}{\left(v^{2}+16v+63\right)^{2}}$$