Řešení - Derivative

Expanding the derivative for multiplication.

$$\frac{d}{dx}\left[3x^{25}x\right]=\frac{d}{dx}\left[3\right]\times x^{25}x+3\times \frac{d}{dx}\left[x^{25}\right]\times x+3x^{25}\times \frac{d}{dx}\left[x\right]$$

Expanding the derivative for multiplication.

$$\frac{d}{dx}\left[3x^{25}x\right]=\frac{d}{dx}\left[3\right]\times x^{25}x+3\times \frac{d}{dx}\left[x^{25}\right]\times x+3x^{25}\times \frac{d}{dx}\left[x\right]$$

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

$$\frac{d}{dx}\left[3x^{25}x\right]=\frac{d}{dx}[3\times \left(x^{25}x\right)]$$

Applying the product rule of derivatives.

$$\frac{d}{dx}[3\times \left(x^{25}x\right)]=\frac{d}{dx}\left[3\right]\times \left(x^{25}x\right)+3\times \frac{d}{dx}\left[x^{25}x\right]$$

Expanding the derivative for multiplication.

$$\frac{d}{dx}\left[3x^{25}x\right]=\frac{d}{dx}\left[3\right]\times x^{25}x+3\times \frac{d}{dx}\left[x^{25}\right]\times x+3x^{25}\times \frac{d}{dx}\left[x\right]$$

Applying the product rule of derivatives.

$$\frac{d}{dx}\left[x^{25}x\right]=\frac{d}{dx}\left[x^{25}\right]\times x+x^{25}\times \frac{d}{dx}\left[x\right]$$

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

$$\frac{d}{dx}\left[3\right]\times \left(x^{25}x\right)+3\left(\frac{d}{dx}\left[x^{25}\right]\times x+x^{25}\times \frac{d}{dx}\left[x\right]\right)=\frac{d}{dx}\left[3\right]\times x^{25}x+3\left(\frac{d}{dx}\left[x^{25}\right]\times x+x^{25}\times \frac{d}{dx}\left[x\right]\right)$$

Multiplying a number by a sum or difference of two numbers can be done by multiplying each number individually and then adding or subtracting the results.

$$\frac{d}{dx}\left[3\right]\times x^{25}x+3\left(\frac{d}{dx}\left[x^{25}\right]\times x+x^{25}\times \frac{d}{dx}\left[x\right]\right)=\frac{d}{dx}\left[3\right]\times x^{25}x+\left(3\times \left(\frac{d}{dx}\left[x^{25}\right]\times x\right)+3\times \left(x^{25}\times \frac{d}{dx}\left[x\right]\right)\right)$$

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

$$\frac{d}{dx}\left[3\right]\times x^{25}x+\left(3\times \left(\frac{d}{dx}\left[x^{25}\right]\times x\right)+3\times \left(x^{25}\times \frac{d}{dx}\left[x\right]\right)\right)=\frac{d}{dx}\left[3\right]\times x^{25}x+\left(3\times \frac{d}{dx}\left[x^{25}\right]\times x+3\times \left(x^{25}\times \frac{d}{dx}\left[x\right]\right)\right)$$

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

$$\frac{d}{dx}\left[3\right]\times x^{25}x+\left(3\times \frac{d}{dx}\left[x^{25}\right]\times x+3\times \left(x^{25}\times \frac{d}{dx}\left[x\right]\right)\right)=\frac{d}{dx}\left[3\right]\times x^{25}x+\left(3\times \frac{d}{dx}\left[x^{25}\right]\times x+3x^{25}\times \frac{d}{dx}\left[x\right]\right)$$

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

$$\frac{d}{dx}\left[3\right]\times x^{25}x+\left(3\times \frac{d}{dx}\left[x^{25}\right]\times x+3x^{25}\times \frac{d}{dx}\left[x\right]\right)=\frac{d}{dx}\left[3\right]\times x^{25}x+3\times \frac{d}{dx}\left[x^{25}\right]\times x+3x^{25}\times \frac{d}{dx}\left[x\right]$$