Riješi x/x^2+10x+24-4/x^2+6x+8

$$\frac { x } { x ^ { 2 } + 10 x + 24 } - \frac { 4 } { x ^ { 2 } + 6 x + 8 }$$

Izračunaj

$$\frac{x-6}{\left(x+2\right)\left(x+6\right)}$$

Kratki koraci za rješenje

Rastavite x^{2}+10x+24 na faktore. Rastavite x^{2}+6x+8 na faktore.

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

Da biste zbrojili ili oduzeli izraze, proširite ih da bi imali iste nazivnike. Najmanji zajednički višekratnik brojeva \left(x+4\right)\left(x+6\right) i \left(x+2\right)\left(x+4\right) jest \left(x+2\right)\left(x+4\right)\left(x+6\right). Pomnožite \frac{x}{\left(x+4\right)\left(x+6\right)} i \frac{x+2}{x+2}. Pomnožite \frac{4}{\left(x+2\right)\left(x+4\right)} i \frac{x+6}{x+6}.

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

Budući da \frac{x\left(x+2\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} i \frac{4\left(x+6\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} imaju isti nazivnik, oduzmite ih oduzimanje njihovih brojnika.

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

Pomnožite izraz x\left(x+2\right)-4\left(x+6\right).

$$\frac{x^{2}+2x-4x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}$$

Kombinirajte slične izraze u x^{2}+2x-4x-24.

$$\frac{x^{2}-2x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}$$

Rastavite na faktore izraze koji još nisu rastavljeni na faktore u izrazu \frac{x^{2}-2x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}.

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

Skratite x+4 u brojniku i nazivniku.

$$\frac{x-6}{\left(x+2\right)\left(x+6\right)}$$

Proširivanje broja \left(x+2\right)\left(x+6\right).

$$\frac{x-6}{x^{2}+8x+12}$$

Diferenciraj u odnosu na x

$$\frac{60+12x-x^{2}}{x^{4}+16x^{3}+88x^{2}+192x+144}$$

Koraci korištenja pravila deriviranja za kvocijent

Rastavite x^{2}+10x+24 na faktore. Rastavite x^{2}+6x+8 na faktore.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{\left(x+4\right)\left(x+6\right)}-\frac{4}{\left(x+2\right)\left(x+4\right)})$$

Da biste zbrojili ili oduzeli izraze, proširite ih da bi imali iste nazivnike. Najmanji zajednički višekratnik brojeva \left(x+4\right)\left(x+6\right) i \left(x+2\right)\left(x+4\right) jest \left(x+2\right)\left(x+4\right)\left(x+6\right). Pomnožite \frac{x}{\left(x+4\right)\left(x+6\right)} i \frac{x+2}{x+2}. Pomnožite \frac{4}{\left(x+2\right)\left(x+4\right)} i \frac{x+6}{x+6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\left(x+2\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}-\frac{4\left(x+6\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)})$$

Budući da \frac{x\left(x+2\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} i \frac{4\left(x+6\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} imaju isti nazivnik, oduzmite ih oduzimanje njihovih brojnika.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\left(x+2\right)-4\left(x+6\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)})$$

Pomnožite izraz x\left(x+2\right)-4\left(x+6\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+2x-4x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)})$$

Kombinirajte slične izraze u x^{2}+2x-4x-24.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}-2x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)})$$

Rastavite na faktore izraze koji još nisu rastavljeni na faktore u izrazu \frac{x^{2}-2x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-6\right)\left(x+4\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)})$$

Skratite x+4 u brojniku i nazivniku.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x-6}{\left(x+2\right)\left(x+6\right)})$$

Koristite svojstvo distributivnosti da biste pomnožili x+2 s x+6 i kombinirali slične izraze.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x-6}{x^{2}+8x+12})$$

Za svake dvije različite funkcije, derivacija kvocijenta dviju funkcija jednaka je nazivniku pomnoženom s derivacijom brojnika minus brojniku pomnoženom s derivacijom nazivnika, sve podijeljeno nazivnikom na kvadrat.

$$\frac{\left(x^{2}+8x^{1}+12\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-6)-\left(x^{1}-6\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+8x^{1}+12)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Derivacija polinoma zbroj je derivacija njegovih dijelova. Derivacija bilo kojeg konstantnog izraza je 0. Derivacija izraza ax^{n} je nax^{n-1}.

$$\frac{\left(x^{2}+8x^{1}+12\right)x^{1-1}-\left(x^{1}-6\right)\left(2x^{2-1}+8x^{1-1}\right)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Pojednostavnite.

$$\frac{\left(x^{2}+8x^{1}+12\right)x^{0}-\left(x^{1}-6\right)\left(2x^{1}+8x^{0}\right)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Pomnožite x^{2}+8x^{1}+12 i x^{0}.

$$\frac{x^{2}x^{0}+8x^{1}x^{0}+12x^{0}-\left(x^{1}-6\right)\left(2x^{1}+8x^{0}\right)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Pomnožite x^{1}-6 i 2x^{1}+8x^{0}.

$$\frac{x^{2}x^{0}+8x^{1}x^{0}+12x^{0}-\left(x^{1}\times 2x^{1}+x^{1}\times 8x^{0}-6\times 2x^{1}-6\times 8x^{0}\right)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Da biste pomnožili potencije s istom bazom, zbrojite njihove eksponente.

$$\frac{x^{2}+8x^{1}+12x^{0}-\left(2x^{1+1}+8x^{1}-6\times 2x^{1}-6\times 8x^{0}\right)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Pojednostavnite.

$$\frac{x^{2}+8x^{1}+12x^{0}-\left(2x^{2}+8x^{1}-12x^{1}-48x^{0}\right)}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Kombinirajte slične izraze.

$$\frac{-x^{2}+12x^{1}+60x^{0}}{\left(x^{2}+8x^{1}+12\right)^{2}}$$

Za svaki izraz t, t^{1}=t.

$$\frac{-x^{2}+12x+60x^{0}}{\left(x^{2}+8x+12\right)^{2}}$$

Za svaki izraz t osim 0, t^{0}=1.

$$\frac{-x^{2}+12x+60\times 1}{\left(x^{2}+8x+12\right)^{2}}$$

Za svaki izraz t, t\times 1=t i 1t=t.

$$\frac{-x^{2}+12x+60}{\left(x^{2}+8x+12\right)^{2}}$$