Issolvi 3/1+x-2x^2+x/x-1

$$\frac { 3 } { 1 + x - 2 x ^ { 2 } } + \frac { x } { x - 1 }$$

Evalwa

$$\frac{2x+3}{2x+1}$$

Passi Qosra tas-Soluzzjonijiet

Iffattura 1+x-2x^{2}.

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

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' \left(-x+1\right)\left(2x+1\right) u x-1 huwa \left(x-1\right)\left(2x+1\right). Immultiplika \frac{3}{\left(-x+1\right)\left(2x+1\right)} b'\frac{-1}{-1}. Immultiplika \frac{x}{x-1} b'\frac{2x+1}{2x+1}.

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

Billi \frac{3\left(-1\right)}{\left(x-1\right)\left(2x+1\right)} u \frac{x\left(2x+1\right)}{\left(x-1\right)\left(2x+1\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-multiplikazzjonijiet fi 3\left(-1\right)+x\left(2x+1\right).

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

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

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

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

$$\frac{2x+3}{2x+1}$$

Iddifferenzja w.r.t. x

$$-\frac{4}{\left(2x+1\right)^{2}}$$

Passi għall-Użu ta' Regola Derivattiva għal Kwozjent

Iffattura 1+x-2x^{2}.

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

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' \left(-x+1\right)\left(2x+1\right) u x-1 huwa \left(x-1\right)\left(2x+1\right). Immultiplika \frac{3}{\left(-x+1\right)\left(2x+1\right)} b'\frac{-1}{-1}. Immultiplika \frac{x}{x-1} b'\frac{2x+1}{2x+1}.

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

Billi \frac{3\left(-1\right)}{\left(x-1\right)\left(2x+1\right)} u \frac{x\left(2x+1\right)}{\left(x-1\right)\left(2x+1\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-multiplikazzjonijiet fi 3\left(-1\right)+x\left(2x+1\right).

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

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+3}{2x+1})$$

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.

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

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}.

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

Agħmel l-aritmetika.

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

Espandi bl-użu ta' propjetà distributtiva.

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

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

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

Agħmel l-aritmetika.

$$\frac{4x^{1}+2x^{0}-\left(4x^{1}+6x^{0}\right)}{\left(2x^{1}+1\right)^{2}}$$

Neħħi l-parenteżi mhux meħtieġa.

$$\frac{4x^{1}+2x^{0}-4x^{1}-6x^{0}}{\left(2x^{1}+1\right)^{2}}$$

Ikkombina termini simili.

$$\frac{\left(4-4\right)x^{1}+\left(2-6\right)x^{0}}{\left(2x^{1}+1\right)^{2}}$$

Naqqas 4 minn 4 u 6 minn 2.

$$\frac{-4x^{0}}{\left(2x^{1}+1\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{-4x^{0}}{\left(2x+1\right)^{2}}$$

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

$$\frac{-4}{\left(2x+1\right)^{2}}$$