Řešení - Derivative

Computing the derivative of a tangent function using the chain rule.

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

Decomposing the function for the chain rule.

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

Computing the derivative of a tangent function using the chain rule.

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

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

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

Computing the derivative of a fraction.

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

Computing the derivative of a sine function.

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

Computing the derivative of a cosine function.

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

Canceling out two minus signs, which results in a positive value.

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

Multiplying the same numbers together.

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

Multiplying the same numbers together.

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

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

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

Substituting the variable back into the function.

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