Řešení - Derivative

Computing the derivative of a cotangent function.

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

The cotangent of an angle is equal to the cosine of the angle divided by the sine of the angle.

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

Computing the derivative of a fraction.

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

Computing the derivative of a cosine function.

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

Computing the derivative of a sine function.

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

Multiplying the same numbers together.

$$\frac{-\sin \left(x\right)\times \sin \left(x\right)-\cos \left(x\right)\times \cos \left(x\right)}{\sin {\left(x\right)}^2}=\frac{-\sin {\left(x\right)}^2-\cos \left(x\right)\times \cos \left(x\right)}{\sin {\left(x\right)}^2}$$

Multiplying the same numbers together.

$$\frac{-\sin {\left(x\right)}^2-\cos \left(x\right)\times \cos \left(x\right)}{\sin {\left(x\right)}^2}=\frac{-\sin {\left(x\right)}^2-\cos {\left(x\right)}^2}{\sin {\left(x\right)}^2}$$

Grouping negative terms together.

$$\frac{-\sin {\left(x\right)}^2-\cos {\left(x\right)}^2}{\sin {\left(x\right)}^2}=\frac{-\left(\sin {\left(x\right)}^2+\cos {\left(x\right)}^2\right)}{\sin {\left(x\right)}^2}$$

The square of the sine of an angle plus the square of the cosine of the same angle is always equal to one.

$$\frac{-\left(\sin {\left(x\right)}^2+\cos {\left(x\right)}^2\right)}{\sin {\left(x\right)}^2}=\frac{-1}{\sin {\left(x\right)}^2}$$

Placing the minus sign in front of a fraction.

$$\frac{-1}{\sin {\left(x\right)}^2}=-\frac{1}{\sin {\left(x\right)}^2}$$

Computing the derivative of a cosecant function.

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

The cosecant of an angle is equal to one divided by the sine of the angle.

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

Computing the derivative of a reciprocal.

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

Computing the derivative of a sine function.

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