Soluție - Derivată

Derivata unei valori constante este întotdeauna zero.

$$\frac{d}{dx}\left[4\right]\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \frac{d}{dx}\left[{\left(2x^2+5\right)}^3\right]\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \frac{d}{dx}\left[{\left(2x^2+5\right)}^3\right]\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]$$

Calcularea derivatei unei funcții de putere.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \frac{d}{dx}\left[{\left(2x^2+5\right)}^3\right]\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]$$

Aplicarea regulii produsului derivatelor.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[4\right]\times x+4\times \frac{d}{dx}\left[x\right]\right)$$

Derivata unei valori constante este întotdeauna zero.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[4\right]\times x+4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0x+4\times \frac{d}{dx}\left[x\right]\right)$$

Înmulțirea unui număr cu zero întotdeauna rezultă în zero.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0x+4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0+4\times \frac{d}{dx}\left[x\right]\right)$$

Adăugând zero la un număr, care nu îi schimbă valoarea.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0+4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times \frac{d}{dx}\left[x\right]\right)$$

Derivata unei variabile în raport cu ea însăși este întotdeauna egală cu unu.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times 1\right)$$

Înmulțind un număr cu unu, care nu-i schimbă valoarea.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times 1\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Derivata unei valori constante este întotdeauna zero.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Aplicarea regulii sumei derivatelor.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Înmulțirea unui număr cu zero întotdeauna rezultă în zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Aplicarea regulii produsului derivatelor.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(\frac{d}{dx}\left[2\right]\times x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Derivata unei valori constante este întotdeauna zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(\frac{d}{dx}\left[2\right]\times x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Înmulțirea unui număr cu zero întotdeauna rezultă în zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Adăugând zero la un număr, care nu îi schimbă valoarea.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Calcularea derivatei x ridicat la puterea n.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^{2-1}\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Scăderea valorii unu dintr-un număr.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^{2-1}\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^1\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Înmulțirea poate fi grupată în mod diferit, dar rezultatul rămâne același.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^1\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(2\times 2\right)\times x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Înmulțirea a două numere întregi împreună.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(2\times 2\right)\times x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(4x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Derivata unei valori constante este întotdeauna zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(4x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Adăugând zero la un număr, care nu îi schimbă valoarea.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\times \left(0+\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4$$

Simplificarea expresiilor aritmetice.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4$$

Adăugând zero la un număr, care nu îi schimbă valoarea.

$$0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4$$

Simplificarea expresiilor aritmetice.

$$0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Adăugând zero la un număr, care nu îi schimbă valoarea.

$$0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificarea expresiilor aritmetice.

$$4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificarea expresiilor aritmetice.

$$4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{12x}{2x^2+5}\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificarea expresiilor aritmetice.

$$4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{12x}{2x^2+5}\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left(12x{\left(2x^2+5\right)}^2\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificarea expresiilor aritmetice.

$$4\times \left(12x{\left(2x^2+5\right)}^2\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=192x^2{\left(2x^2+5\right)}^2+16{\left(2x^2+5\right)}^3$$