คำตอบ - Derivative

Expanding the derivative for multiplication.

$$f\times \frac{d}{dx}[2f\times \left(3fx\right)]=f\left(\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)$$

Expanding the derivative for multiplication.

$$f\times \frac{d}{dx}[2f\times \left(3fx\right)]=f\left(\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)$$

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

$$\frac{d}{dx}[2f\times \left(3fx\right)]=\frac{d}{dx}[2\times \left(f\times \left(3fx\right)\right)]$$

Applying the product rule of derivatives.

$$\frac{d}{dx}[2\times \left(f\times \left(3fx\right)\right)]=\frac{d}{dx}\left[2\right]\times \left(f\times \left(3fx\right)\right)+2\times \frac{d}{dx}[f\times \left(3fx\right)]$$

Expanding the derivative for multiplication.

$$f\times \frac{d}{dx}[2f\times \left(3fx\right)]=f\left(\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)$$

Applying the product rule of derivatives.

$$\frac{d}{dx}[f\times \left(3fx\right)]=\frac{d}{dx}\left[f\right]\times \left(3fx\right)+f\times \frac{d}{dx}\left[3fx\right]$$

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

$$\frac{d}{dx}\left[2\right]\times \left(f\times \left(3fx\right)\right)+2\left(\frac{d}{dx}\left[f\right]\times \left(3fx\right)+f\times \frac{d}{dx}\left[3fx\right]\right)=\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+2\left(\frac{d}{dx}\left[f\right]\times \left(3fx\right)+f\times \frac{d}{dx}\left[3fx\right]\right)$$

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.

$$\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+2\left(\frac{d}{dx}\left[f\right]\times \left(3fx\right)+f\times \frac{d}{dx}\left[3fx\right]\right)=\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+\left(2\times \left(\frac{d}{dx}\left[f\right]\times \left(3fx\right)\right)+2\times \left(f\times \frac{d}{dx}\left[3fx\right]\right)\right)$$

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

$$\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+\left(2\times \left(\frac{d}{dx}\left[f\right]\times \left(3fx\right)\right)+2\times \left(f\times \frac{d}{dx}\left[3fx\right]\right)\right)=\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+\left(2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2\times \left(f\times \frac{d}{dx}\left[3fx\right]\right)\right)$$

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

$$\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+\left(2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2\times \left(f\times \frac{d}{dx}\left[3fx\right]\right)\right)=\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+\left(2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)$$

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

$$\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+\left(2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)=\frac{d}{dx}\left[2\right]\times f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]$$

Expanding the derivative for multiplication.

$$f\left(0f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)=f\left(0f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\left(\frac{d}{dx}\left[3\right]\times fx+3\times \frac{d}{dx}\left[f\right]\times x+3f\times \frac{d}{dx}\left[x\right]\right)\right)$$

Expanding the derivative for multiplication.

$$f\left(0f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)=f\left(0f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\left(\frac{d}{dx}\left[3\right]\times fx+3\times \frac{d}{dx}\left[f\right]\times x+3f\times \frac{d}{dx}\left[x\right]\right)\right)$$

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

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

Applying the product rule of derivatives.

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

Expanding the derivative for multiplication.

$$f\left(0f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\times \frac{d}{dx}\left[3fx\right]\right)=f\left(0f\times \left(3fx\right)+2\times \frac{d}{dx}\left[f\right]\times \left(3fx\right)+2f\left(\frac{d}{dx}\left[3\right]\times fx+3\times \frac{d}{dx}\left[f\right]\times x+3f\times \frac{d}{dx}\left[x\right]\right)\right)$$

Applying the product rule of derivatives.

$$\frac{d}{dx}\left[fx\right]=\frac{d}{dx}\left[f\right]\times x+f\times \frac{d}{dx}\left[x\right]$$

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

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

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.

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

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

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

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

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

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

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