Issolvi x^3+1/x+3

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

Evalwa

$$\frac{x^{4}+3x^{3}+1}{x+3}$$

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika x^{3} b'\frac{x+3}{x+3}.

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

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

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

Agħmel il-multiplikazzjonijiet fi x^{3}\left(x+3\right)+1.

$$\frac{x^{4}+3x^{3}+1}{x+3}$$

Iddifferenzja w.r.t. x

$$\frac{3x^{4}+18x^{3}+27x^{2}-1}{\left(x+3\right)^{2}}$$

Passi għall-Użu ta' Regola Derivattiva għal Kwozjent

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika x^{3} b'\frac{x+3}{x+3}.

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

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

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

Agħmel il-multiplikazzjonijiet fi x^{3}\left(x+3\right)+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{4}+3x^{3}+1}{x+3})$$

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.

$$\frac{\left(x^{1}+3\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+3x^{3}+1)-\left(x^{4}+3x^{3}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}+3)}{\left(x^{1}+3\right)^{2}}$$

Id-derivattiv ta' polynomial huwa s-somma tad-derivattivi tat-termini tiegħu. Id-derivattiv ta' kwalunkwe terminu kostanti huwa 0. Id-derivattiv ta' ax^{n} huwa nax^{n-1}.

$$\frac{\left(x^{1}+3\right)\left(4x^{4-1}+3\times 3x^{3-1}\right)-\left(x^{4}+3x^{3}+1\right)x^{1-1}}{\left(x^{1}+3\right)^{2}}$$

Issimplifika.

$$\frac{\left(x^{1}+3\right)\left(4x^{3}+9x^{2}\right)-\left(x^{4}+3x^{3}+1\right)x^{0}}{\left(x^{1}+3\right)^{2}}$$

Immultiplika x^{1}+3 b'4x^{3}+9x^{2}.

$$\frac{x^{1}\times 4x^{3}+x^{1}\times 9x^{2}+3\times 4x^{3}+3\times 9x^{2}-\left(x^{4}+3x^{3}+1\right)x^{0}}{\left(x^{1}+3\right)^{2}}$$

Immultiplika x^{4}+3x^{3}+1 b'x^{0}.

$$\frac{x^{1}\times 4x^{3}+x^{1}\times 9x^{2}+3\times 4x^{3}+3\times 9x^{2}-\left(x^{4}x^{0}+3x^{3}x^{0}+x^{0}\right)}{\left(x^{1}+3\right)^{2}}$$

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

$$\frac{4x^{1+3}+9x^{1+2}+3\times 4x^{3}+3\times 9x^{2}-\left(x^{4}+3x^{3}+x^{0}\right)}{\left(x^{1}+3\right)^{2}}$$

Issimplifika.

$$\frac{4x^{4}+9x^{3}+12x^{3}+27x^{2}-\left(x^{4}+3x^{3}+x^{0}\right)}{\left(x^{1}+3\right)^{2}}$$

Ikkombina termini simili.

$$\frac{3x^{4}+6x^{3}+12x^{3}+27x^{2}-x^{0}}{\left(x^{1}+3\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{3x^{4}+6x^{3}+12x^{3}+27x^{2}-x^{0}}{\left(x+3\right)^{2}}$$

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

$$\frac{3x^{4}+6x^{3}+12x^{3}+27x^{2}-1}{\left(x+3\right)^{2}}$$