Řešení - Derivative

Applying the sum rule of derivatives.

$$\frac{d}{dx}[x^5+3x^4]=\frac{d}{dx}\left[x^5\right]+\frac{d}{dx}\left[3x^4\right]$$

Computing the derivative of x raised to the power of n.

$$\frac{d}{dx}\left[x^5\right]+\frac{d}{dx}\left[3x^4\right]=5x^{5-1}+\frac{d}{dx}\left[3x^4\right]$$

Subtracting one from a number.

$$5x^{5-1}+\frac{d}{dx}\left[3x^4\right]=5x^4+\frac{d}{dx}\left[3x^4\right]$$

Applying the product rule of derivatives.

$$5x^4+\frac{d}{dx}\left[3x^4\right]=5x^4+\left(\frac{d}{dx}\left[3\right]\times x^4+3\times \frac{d}{dx}\left[x^4\right]\right)$$

The derivative of a constant value is always zero.

$$5x^4+\left(\frac{d}{dx}\left[3\right]\times x^4+3\times \frac{d}{dx}\left[x^4\right]\right)=5x^4+\left(0x^4+3\times \frac{d}{dx}\left[x^4\right]\right)$$

Multiplying a number by zero always results in zero.

$$5x^4+\left(0x^4+3\times \frac{d}{dx}\left[x^4\right]\right)=5x^4+\left(0+3\times \frac{d}{dx}\left[x^4\right]\right)$$

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

$$5x^4+\left(0+3\times \frac{d}{dx}\left[x^4\right]\right)=5x^4+3\times \frac{d}{dx}\left[x^4\right]$$

Computing the derivative of x raised to the power of n.

$$5x^4+3\times \frac{d}{dx}\left[x^4\right]=5x^4+3\times \left(4x^{4-1}\right)$$

Subtracting one from a number.

$$5x^4+3\times \left(4x^{4-1}\right)=5x^4+3\times \left(4x^3\right)$$

Multiplication can be grouped differently, but the result remains the same.

$$5x^4+3\times \left(4x^3\right)=5x^4+\left(3\times 4\right)\times x^3$$

Multiplying two integers together.

$$5x^4+\left(3\times 4\right)\times x^3=5x^4+12x^3$$