Rešenje - Izvodi

Računanje izvoda razlomka.

$$\frac{d}{dx}\left[\frac{3x+5}{x^2}\right]=\frac{\frac{d}{dx}[3x+5]\times x^2-\left(3x+5\right)\times \frac{d}{dx}\left[x^2\right]}{{\left(x^2\right)}^2}$$

Primena pravila zbirke izvoda.

$$\frac{\frac{d}{dx}[3x+5]\times x^2-\left(3x+5\right)\times \frac{d}{dx}\left[x^2\right]}{{\left(x^2\right)}^2}=\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \frac{d}{dx}\left[x^2\right]}{{\left(x^2\right)}^2}$$

Računanje izvoda x na n-ti stepen.

$$\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \frac{d}{dx}\left[x^2\right]}{{\left(x^2\right)}^2}=\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x^{2-1}\right)}{{\left(x^2\right)}^2}$$

Oduzimanje jedinice od broja.

$$\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x^{2-1}\right)}{{\left(x^2\right)}^2}=\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x^1\right)}{{\left(x^2\right)}^2}$$

Bilo koji broj na stepen jedan je jednak tom broju.

$$\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x^1\right)}{{\left(x^2\right)}^2}=\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Primena pravila proizvoda izvoda.

$$\frac{\left(\frac{d}{dx}\left[3x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(\left(\frac{d}{dx}\left[3\right]\times x+3\times \frac{d}{dx}\left[x\right]\right)+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Derivacija konstantne vrednosti je uvek nula.

$$\frac{\left(\left(\frac{d}{dx}\left[3\right]\times x+3\times \frac{d}{dx}\left[x\right]\right)+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(\left(0x+3\times \frac{d}{dx}\left[x\right]\right)+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Množenje broja sa nulom uvek rezultuje nulom.

$$\frac{\left(\left(0x+3\times \frac{d}{dx}\left[x\right]\right)+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(\left(0+3\times \frac{d}{dx}\left[x\right]\right)+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Sabiranje nule sa brojem, što ne menja njegovu vrednost.

$$\frac{\left(\left(0+3\times \frac{d}{dx}\left[x\right]\right)+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(3\times \frac{d}{dx}\left[x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Derivacija promenljive u odnosu na samu sebe uvek je jednaka jedinici.

$$\frac{\left(3\times \frac{d}{dx}\left[x\right]+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(3\times 1+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Množenje broja sa jedinicom, što ne menja njegovu vrednost.

$$\frac{\left(3\times 1+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(3+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Derivacija konstantne vrednosti je uvek nula.

$$\frac{\left(3+\frac{d}{dx}\left[5\right]\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(3+0\right)x^2-\left(3x+5\right)\times \left(2x\right)}{{\left(x^2\right)}^2}$$

Simplifikacija aritmetičkih izraza.

$$\frac{\left(3+0\right)x^2-3x+5\times \left(2x\right)}{{\left(x^2\right)}^2}=\frac{\left(3+0\right)x^2-3x+5\times \left(2x\right)}{x^4}$$

Sabiranje nule sa brojem, što ne menja njegovu vrednost.

$$\frac{\left(3+0\right)x^2-3x+5\times \left(2x\right)}{x^4}=\frac{3x^2-3x+5\times \left(2x\right)}{x^4}$$

Simplifikacija aritmetičkih izraza.

$$\frac{3x^2+-3x+5\times \left(2x\right)}{x^4}=\frac{3x^2+-3x-5\times \left(2x\right)}{x^4}$$

Simplifikacija aritmetičkih izraza.

$$\frac{3x^2-3x-5\times \left(2x\right)}{x^4}=\frac{3x^2+2x\left(-3x-5\right)}{x^4}$$