Řešení - Derivative

Expanding the derivative for multiplication.

$$\frac{d}{dx}\left[2xy\right]=\frac{d}{dx}\left[2\right]\times xy+2\times \frac{d}{dx}\left[x\right]\times y+2x\times \frac{d}{dx}\left[y\right]$$

Expanding the derivative for multiplication.

$$\frac{d}{dx}\left[2xy\right]=\frac{d}{dx}\left[2\right]\times xy+2\times \frac{d}{dx}\left[x\right]\times y+2x\times \frac{d}{dx}\left[y\right]$$

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

$$\frac{d}{dx}\left[2xy\right]=\frac{d}{dx}[2\times \left(xy\right)]$$

Applying the product rule of derivatives.

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

Expanding the derivative for multiplication.

$$\frac{d}{dx}\left[2xy\right]=\frac{d}{dx}\left[2\right]\times xy+2\times \frac{d}{dx}\left[x\right]\times y+2x\times \frac{d}{dx}\left[y\right]$$

Applying the product rule of derivatives.

$$\frac{d}{dx}\left[xy\right]=\frac{d}{dx}\left[x\right]\times y+x\times \frac{d}{dx}\left[y\right]$$

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

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

Multiplying a number by a sum or difference of two numbers can be done by multiplying each number individually and then adding or subtracting the results.

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

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

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

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

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

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

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