Řešení - Derivative

Applying the product rule of derivatives.

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

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

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

Subtracting one from a number.

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

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

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

Computing the derivative of a power function.

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

The derivative of a constant value is always zero.

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

Simplifying the arithmetic expressions.

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

Multiplying a number by zero always results in zero.

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

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

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