Řešení - Derivative

Applying the sum rule of derivatives.

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

Applying the product rule of derivatives.

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

The derivative of a constant value is always zero.

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

Multiplying a number by zero always results in zero.

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

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

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

The derivative of a variable with respect to itself is always equal to one.

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

Multiplying a number by one, which does not change its value.

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

Applying the product rule of derivatives.

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

The derivative of a constant value is always zero.

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

Multiplying a number by zero always results in zero.

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

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

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

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

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

Subtracting one from a number.

$$2+3\times \left(2x^{2-1}\right)=2+3\times \left(2x^1\right)$$

Multiplication can be grouped differently, but the result remains the same.

$$2+3\times \left(2x^1\right)=2+\left(3\times 2\right)\times x^1$$

Multiplying two integers together.

$$2+\left(3\times 2\right)\times x^1=2+6x^1$$

Simplifying the arithmetic expressions.

$$2+6x^1=2+6x$$

Simplifying the arithmetic expressions.

$$2+6x=6x+2$$