Řešení - Derivative

Computing the derivative of a fraction.

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

Computing the derivative of a power function.

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

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

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

The derivative of a constant value is always zero.

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

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

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

Multiplying a number by zero always results in zero.

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

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

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