Řešení - Derivative

Expanding the derivative for multiplication.

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

Expanding the derivative for multiplication.

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

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

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

Applying the product rule of derivatives.

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

Expanding the derivative for multiplication.

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

Expanding the derivative for multiplication.

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

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

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

Applying the product rule of derivatives.

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

Expanding the derivative for multiplication.

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

Applying the product rule of derivatives.

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

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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