Řešení - Derivative

Applying the product rule of derivatives.

$$\frac{d}{dx}[-1\times \ln \left(\sin \left(x\right)\right)]=\frac{d}{dx}[-1]\times \ln \left(\sin \left(x\right)\right)-1\times \frac{d}{dx}\left[\ln \left(\sin \left(x\right)\right)\right]$$

The derivative of a constant value is always zero.

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

Multiplying a number by zero always results in zero.

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

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

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

Computing the derivative of a sine function.

$$-1\times \left(\frac{1}{\sin \left(x\right)}\times \frac{d}{dx}\left[\sin \left(x\right)\right]\right)=-1\times \left(\frac{1}{\sin \left(x\right)}\times \cos \left(x\right)\right)$$

Simplifying the arithmetic expressions.

$$-1\times \left(\frac{1}{\sin \left(x\right)}\times \cos \left(x\right)\right)=-1\times \left(\frac{\cos \left(x\right)}{\sin \left(x\right)}\right)$$

Simplifying the arithmetic expressions.

$$-1\times \left(\frac{\cos \left(x\right)}{\sin \left(x\right)}\right)=-\frac{\cos \left(x\right)}{\sin \left(x\right)}$$