Rozwiązanie - Pochodna

Pochodna stałej wartości zawsze wynosi zero.

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

Obliczanie pochodnej funkcji potęgowej.

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

Zastosowanie reguły iloczynu pochodnych.

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

Pochodna stałej wartości zawsze wynosi zero.

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

Mnożenie liczby przez zero zawsze daje wynik zero.

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

Dodawanie zera do liczby, co nie zmienia jej wartości.

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

Pochodna zmiennej względem samej siebie zawsze wynosi jeden.

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

Mnożenie liczby przez jeden, co nie zmienia jej wartości.

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

Pochodna stałej wartości zawsze wynosi zero.

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

Zastosowanie reguły sumy pochodnych.

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

Mnożenie liczby przez zero zawsze daje wynik zero.

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

Zastosowanie reguły iloczynu pochodnych.

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

Pochodna stałej wartości zawsze wynosi zero.

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

Mnożenie liczby przez zero zawsze daje wynik zero.

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

Dodawanie zera do liczby, co nie zmienia jej wartości.

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

Obliczanie pochodnej dla x o potędze n.

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

Odejmowanie jedynki od liczby.

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

Mnożenie może być grupowane w różny sposób, ale wynik pozostaje taki sam.

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

Mnożenie dwóch liczb całkowitych.

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

Pochodna stałej wartości zawsze wynosi zero.

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

Dodawanie zera do liczby, co nie zmienia jej wartości.

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

Uproszczanie wyrażeń arytmetycznych.

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

Dodawanie zera do liczby, co nie zmienia jej wartości.

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

Uproszczanie wyrażeń arytmetycznych.

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

Dodawanie zera do liczby, co nie zmienia jej wartości.

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

Uproszczanie wyrażeń arytmetycznych.

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

Uproszczanie wyrażeń arytmetycznych.

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

Uproszczanie wyrażeń arytmetycznych.

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

Uproszczanie wyrażeń arytmetycznych.

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