Solution - Derivative

Applying the product rule of derivatives.

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

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

$$\frac{d}{dx}\left[x\right]\times e^{5x}+x\times \frac{d}{dx}\left[e^{5x}\right]=1\times e^{5x}+x\times \frac{d}{dx}\left[e^{5x}\right]$$

Computing the derivative of a power function.

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

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

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

Applying the product rule of derivatives.

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

The derivative of a constant value is always zero.

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

Multiplying a number by zero always results in zero.

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

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

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

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

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

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

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

The derivative of a constant value is always zero.

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

Simplifying the arithmetic expressions.

$$e^{5x}+x\times \left(e^{5x}\times \left(5\times \ln \left(e\right)+\frac{5x}{e}\times 0\right)\right)=e^{5x}+x\times \left(e^{5x}\times \left(5\times 1+\frac{5x}{e}\times 0\right)\right)$$

Multiplying a number by zero always results in zero.

$$e^{5x}+x\times \left(e^{5x}\times \left(5\times 1+\frac{5x}{e}\times 0\right)\right)=e^{5x}+x\times \left(e^{5x}\times \left(5\times 1+0\right)\right)$$

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

$$e^{5x}+x\times \left(e^{5x}\times \left(5\times 1+0\right)\right)=e^{5x}+x\times \left(e^{5x}\times \left(5\times 1\right)\right)$$

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

$$e^{5x}+x\times \left(e^{5x}\times \left(5\times 1\right)\right)=e^{5x}+x\times \left(e^{5x}\times 5\right)$$

Simplifying the arithmetic expressions.

$$e^{5x}+x\times \left(e^{5x}\times 5\right)=e^{5x}+x\times \left(5e^{5x}\right)$$

Simplifying the arithmetic expressions.

$$e^{5x}+x\times \left(5e^{5x}\right)=e^{5x}+5x{e}^{5x}$$

Simplifying the arithmetic expressions.

$$e^{5x}+5x{e}^{5x}=5x{e}^{5x}+e^{5x}$$