Issolvi 3/1-x+1/1+x-28/1-x^2

$$\frac { 3 } { 1 - x } + \frac { 1 } { 1 + x } - \frac { 28 } { 1 - x ^ { 2 } }$$

Evalwa

$$\frac{2\left(x-12\right)}{1-x^{2}}$$

Passi Qosra tas-Soluzzjonijiet

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

$$\frac{3\left(x+1\right)}{\left(x+1\right)\left(-x+1\right)}+\frac{-x+1}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}}$$

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

$$\frac{3\left(x+1\right)-x+1}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}}$$

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

$$\frac{3x+3-x+1}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}}$$

Ikkombina termini simili f'3x+3-x+1.

$$\frac{2x+4}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}}$$

Iffattura 1-x^{2}.

$$\frac{2x+4}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{\left(x-1\right)\left(-x-1\right)}$$

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' \left(x+1\right)\left(-x+1\right) u \left(x-1\right)\left(-x-1\right) huwa \left(x-1\right)\left(x+1\right). Immultiplika \frac{2x+4}{\left(x+1\right)\left(-x+1\right)} b'\frac{-1}{-1}. Immultiplika \frac{28}{\left(x-1\right)\left(-x-1\right)} b'\frac{-1}{-1}.

$$\frac{-\left(2x+4\right)}{\left(x-1\right)\left(x+1\right)}-\frac{28\left(-1\right)}{\left(x-1\right)\left(x+1\right)}$$

Billi \frac{-\left(2x+4\right)}{\left(x-1\right)\left(x+1\right)} u \frac{28\left(-1\right)}{\left(x-1\right)\left(x+1\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{-\left(2x+4\right)-28\left(-1\right)}{\left(x-1\right)\left(x+1\right)}$$

Agħmel il-multiplikazzjonijiet fi -\left(2x+4\right)-28\left(-1\right).

$$\frac{-2x-4+28}{\left(x-1\right)\left(x+1\right)}$$

Ikkombina termini simili f'-2x-4+28.

$$\frac{-2x+24}{\left(x-1\right)\left(x+1\right)}$$

Espandi \left(x-1\right)\left(x+1\right).

$$\frac{-2x+24}{x^{2}-1}$$

Iddifferenzja w.r.t. x

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

Passi għall-Użu ta' Regola Derivattiva għal Kwozjent

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\left(x+1\right)}{\left(x+1\right)\left(-x+1\right)}+\frac{-x+1}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\left(x+1\right)-x+1}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+3-x+1}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}})$$

Ikkombina termini simili f'3x+3-x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+4}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{1-x^{2}})$$

Iffattura 1-x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+4}{\left(x+1\right)\left(-x+1\right)}-\frac{28}{\left(x-1\right)\left(-x-1\right)})$$

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' \left(x+1\right)\left(-x+1\right) u \left(x-1\right)\left(-x-1\right) huwa \left(x-1\right)\left(x+1\right). Immultiplika \frac{2x+4}{\left(x+1\right)\left(-x+1\right)} b'\frac{-1}{-1}. Immultiplika \frac{28}{\left(x-1\right)\left(-x-1\right)} b'\frac{-1}{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-\left(2x+4\right)}{\left(x-1\right)\left(x+1\right)}-\frac{28\left(-1\right)}{\left(x-1\right)\left(x+1\right)})$$

Billi \frac{-\left(2x+4\right)}{\left(x-1\right)\left(x+1\right)} u \frac{28\left(-1\right)}{\left(x-1\right)\left(x+1\right)} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-\left(2x+4\right)-28\left(-1\right)}{\left(x-1\right)\left(x+1\right)})$$

Agħmel il-multiplikazzjonijiet fi -\left(2x+4\right)-28\left(-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x-4+28}{\left(x-1\right)\left(x+1\right)})$$

Ikkombina termini simili f'-2x-4+28.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x+24}{\left(x-1\right)\left(x+1\right)})$$

Ikkunsidra li \left(x-1\right)\left(x+1\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Ikkwadra 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x+24}{x^{2}-1})$$

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.

$$\frac{\left(x^{2}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(-2x^{1}+24)-\left(-2x^{1}+24\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-1)}{\left(x^{2}-1\right)^{2}}$$

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}.

$$\frac{\left(x^{2}-1\right)\left(-2\right)x^{1-1}-\left(-2x^{1}+24\right)\times 2x^{2-1}}{\left(x^{2}-1\right)^{2}}$$

Agħmel l-aritmetika.

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

Espandi bl-użu ta' propjetà distributtiva.

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

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

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

Agħmel l-aritmetika.

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

Neħħi l-parenteżi mhux meħtieġa.

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

Ikkombina termini simili.

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

Naqqas -4 minn -2.

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

Għal kwalunkwe terminu t, t^{1}=t.

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

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

$$\frac{2x^{2}+2\times 1-48x}{\left(x^{2}-1\right)^{2}}$$

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

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