Issolvi f(x)=x+1/x^2+1

$$f ( x ) = x + \frac { 1 } { x ^ { 2 } + 1 }$$

Evalwa

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

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika x b'\frac{x^{2}+1}{x^{2}+1}.

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

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

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

Agħmel il-multiplikazzjonijiet fi x\left(x^{2}+1\right)+1.

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

Iddifferenzja w.r.t. x

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

Passi għall-Użu ta' Regola Derivattiva għal Kwozjent

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. Immultiplika x b'\frac{x^{2}+1}{x^{2}+1}.

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

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

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

Agħmel il-multiplikazzjonijiet fi x\left(x^{2}+1\right)+1.

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

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.

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

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}.

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

Issimplifika.

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

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

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

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

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

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

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

Issimplifika.

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

Ikkombina termini simili.

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

Għal kwalunkwe terminu t, t^{1}=t.

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

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

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