Řešení - Derivative

Applying the product rule of derivatives.

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

Applying the sum rule of derivatives.

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

Applying the sum rule of derivatives.

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

Computing the derivative of x raised to the power of n.

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

Subtracting one from a number.

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

Any number raised to the power of one equals the number itself.

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

The derivative of a constant value is always zero.

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

Computing the derivative of x raised to the power of n.

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

Subtracting one from a number.

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

The derivative of a constant value is always zero.

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

Adding zero to a number, which does not change its value.

$$\left(2x+0\right)\times \left(x^3+2\right)+x^2+5\times \left(3x^2+0\right)=\left(2x\right)\times \left(x^3+2\right)+x^2+5\times \left(3x^2+0\right)$$

Adding zero to a number, which does not change its value.

$$\left(2x\right)\times \left(x^3+2\right)+x^2+5\times \left(3x^2+0\right)=\left(2x\right)\times \left(x^3+2\right)+x^2+5\times \left(3x^2\right)$$

Simplifying the arithmetic expressions.

$$2x\times \left(x^3+2\right)+x^2+5\times \left(3x^2\right)=2x\left(x^3+2\right)+x^2+5\times \left(3x^2\right)$$

Simplifying the arithmetic expressions.

$$2x\left(x^3+2\right)+x^2+5\times \left(3x^2\right)=2x\left(x^3+2\right)+3x^2\left(x^2+5\right)$$

Simplifying the arithmetic expressions.

$$2x\left(x^3+2\right)+3x^2\left(x^2+5\right)=3x^2\left(x^2+5\right)+2x\left(x^3+2\right)$$