Issolvi 2/x+4+3/x

$$\frac { 2 } { x + 4 } + \frac { 3 } { x }$$

Evalwa

$$\frac{5x+12}{x\left(x+4\right)}$$

Passi Qosra tas-Soluzzjonijiet

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' x+4 u x huwa x\left(x+4\right). Immultiplika \frac{2}{x+4} b'\frac{x}{x}. Immultiplika \frac{3}{x} b'\frac{x+4}{x+4}.

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

Billi \frac{2x}{x\left(x+4\right)} u \frac{3\left(x+4\right)}{x\left(x+4\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-multiplikazzjonijiet fi 2x+3\left(x+4\right).

$$\frac{2x+3x+12}{x\left(x+4\right)}$$

Ikkombina termini simili f'2x+3x+12.

$$\frac{5x+12}{x\left(x+4\right)}$$

Espandi x\left(x+4\right).

$$\frac{5x+12}{x^{2}+4x}$$

Iddifferenzja w.r.t. x

$$-\frac{5x^{2}+24x+48}{\left(x\left(x+4\right)\right)^{2}}$$

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' x+4 u x huwa x\left(x+4\right). Immultiplika \frac{2}{x+4} b'\frac{x}{x}. Immultiplika \frac{3}{x} b'\frac{x+4}{x+4}.

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

Billi \frac{2x}{x\left(x+4\right)} u \frac{3\left(x+4\right)}{x\left(x+4\right)} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Agħmel il-multiplikazzjonijiet fi 2x+3\left(x+4\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+3x+12}{x\left(x+4\right)})$$

Ikkombina termini simili f'2x+3x+12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+12}{x\left(x+4\right)})$$

Uża l-propjetà distributtiva biex timmultiplika x b'x+4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+12}{x^{2}+4x})$$

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.

$$\frac{\left(x^{2}+4x^{1}\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}+12)-\left(5x^{1}+12\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+4x^{1})}{\left(x^{2}+4x^{1}\right)^{2}}$$

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}.

$$\frac{\left(x^{2}+4x^{1}\right)\times 5x^{1-1}-\left(5x^{1}+12\right)\left(2x^{2-1}+4x^{1-1}\right)}{\left(x^{2}+4x^{1}\right)^{2}}$$

Issimplifika.

$$\frac{\left(x^{2}+4x^{1}\right)\times 5x^{0}-\left(5x^{1}+12\right)\left(2x^{1}+4x^{0}\right)}{\left(x^{2}+4x^{1}\right)^{2}}$$

Immultiplika x^{2}+4x^{1} b'5x^{0}.

$$\frac{x^{2}\times 5x^{0}+4x^{1}\times 5x^{0}-\left(5x^{1}+12\right)\left(2x^{1}+4x^{0}\right)}{\left(x^{2}+4x^{1}\right)^{2}}$$

Immultiplika 5x^{1}+12 b'2x^{1}+4x^{0}.

$$\frac{x^{2}\times 5x^{0}+4x^{1}\times 5x^{0}-\left(5x^{1}\times 2x^{1}+5x^{1}\times 4x^{0}+12\times 2x^{1}+12\times 4x^{0}\right)}{\left(x^{2}+4x^{1}\right)^{2}}$$

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

$$\frac{5x^{2}+4\times 5x^{1}-\left(5\times 2x^{1+1}+5\times 4x^{1}+12\times 2x^{1}+12\times 4x^{0}\right)}{\left(x^{2}+4x^{1}\right)^{2}}$$

Issimplifika.

$$\frac{5x^{2}+20x^{1}-\left(10x^{2}+20x^{1}+24x^{1}+48x^{0}\right)}{\left(x^{2}+4x^{1}\right)^{2}}$$

Ikkombina termini simili.

$$\frac{-5x^{2}-24x^{1}-48x^{0}}{\left(x^{2}+4x^{1}\right)^{2}}$$

Għal kwalunkwe terminu t, t^{1}=t.

$$\frac{-5x^{2}-24x-48x^{0}}{\left(x^{2}+4x\right)^{2}}$$

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

$$\frac{-5x^{2}-24x-48}{\left(x^{2}+4x\right)^{2}}$$