Řešení - Derivative

Convert a number from power form to exponential form using natural logarithm.

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

Convert a number from exponential form to power form using natural logarithm.

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

Multiplication can be done in any order, and the result remains the same.

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

Applying the product rule of derivatives.

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

The derivative of a constant value is always zero.

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

Multiplying a number by zero always results in zero.

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

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

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

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

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

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

$$2^x\times \left(\ln \left(2\right)\times 1\right)=2^x\times \ln \left(2\right)$$

Simplifying the arithmetic expressions.

$$2^x\times \ln \left(2\right)=2^x\ln \left(2\right)$$