Rešite x/x^2+10x+24-4/x^2+6x+8

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

Ovrednoti

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

Kratka navodila za rešitev

Faktorizirajte x^{2}+10x+24. Faktorizirajte x^{2}+6x+8.

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

Če želite prišteti ali odšteti izraze, jih razširite na skupne imenovalce. Najmanjši skupni mnogokratnik \left(x+4\right)\left(x+6\right) in \left(x+2\right)\left(x+4\right) je \left(x+2\right)\left(x+4\right)\left(x+6\right). Pomnožite \frac{x}{\left(x+4\right)\left(x+6\right)} s/z \frac{x+2}{x+2}. Pomnožite \frac{4}{\left(x+2\right)\left(x+4\right)} s/z \frac{x+6}{x+6}.

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

Ker \frac{x\left(x+2\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} in \frac{4\left(x+6\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

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

Izvedi množenje v x\left(x+2\right)-4\left(x+6\right).

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

Združite podobne člene v x^{2}+2x-4x-24.

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

Faktorizirajte izraze, ki še niso faktorizirani v \frac{x^{2}-2x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}.

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

Okrajšaj x+4 v števcu in imenovalcu.

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

Razčlenite \left(x+2\right)\left(x+6\right).

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

Odvajajte w.r.t. x

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

Koraki z uporabo pravila odvoda za kvocient

Faktorizirajte x^{2}+10x+24. Faktorizirajte x^{2}+6x+8.

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

Če želite prišteti ali odšteti izraze, jih razširite na skupne imenovalce. Najmanjši skupni mnogokratnik \left(x+4\right)\left(x+6\right) in \left(x+2\right)\left(x+4\right) je \left(x+2\right)\left(x+4\right)\left(x+6\right). Pomnožite \frac{x}{\left(x+4\right)\left(x+6\right)} s/z \frac{x+2}{x+2}. Pomnožite \frac{4}{\left(x+2\right)\left(x+4\right)} s/z \frac{x+6}{x+6}.

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

Ker \frac{x\left(x+2\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} in \frac{4\left(x+6\right)}{\left(x+2\right)\left(x+4\right)\left(x+6\right)} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

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

Izvedi množenje v x\left(x+2\right)-4\left(x+6\right).

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

Združite podobne člene v x^{2}+2x-4x-24.

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

Faktorizirajte izraze, ki še niso faktorizirani v \frac{x^{2}-2x-24}{\left(x+2\right)\left(x+4\right)\left(x+6\right)}.

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

Okrajšaj x+4 v števcu in imenovalcu.

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

Uporabite lastnost distributivnosti za množenje x+2 krat x+6 in kombiniranje pogojev podobnosti.

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

Za kateri koli dve odvedljivi funkciji je odvod kvocienta dveh funkcij imenovalec krat odvod števca minus števec krat odvod imenovalca, vse skupaj pa je deljeno s kvadratom imenovalca.

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

Odvod polinoma je vsota odvodov njegovih členov. Odvod katerega koli prostega člena je 0. Odvod člena ax^{n} je nax^{n-1}.

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

Poenostavite.

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

Pomnožite x^{2}+8x^{1}+12 s/z 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 s/z 2x^{1}+8x^{0}.

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

Če želite množiti potence iste osnove, seštejte njihove eksponente.

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

Poenostavite.

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

Združite podobne člene.

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

Za kakršen koli izraz t, t^{1}=t.

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

Za kakršen koli izraz t, razen 0, t^{0}=1.

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

Za kakršen koli izraz t, t\times 1=t in 1t=t.

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