Derivative of $$$x e^{x}$$$
The calculator will find the derivative of $$$x e^{x}$$$, with steps shown.
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Solution
Apply the product rule $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ with $$$f{\left(x \right)} = x$$$ and $$$g{\left(x \right)} = e^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(x e^{x}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) e^{x} + x \frac{d}{dx} \left(e^{x}\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \frac{d}{dx} \left(e^{x}\right) + e^{x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(e^{x}\right) + e^{x} {\color{red}\left(1\right)}$$The derivative of the exponential is $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} + e^{x} = x {\color{red}\left(e^{x}\right)} + e^{x}$$Simplify:
$$x e^{x} + e^{x} = \left(x + 1\right) e^{x}$$Thus, $$$\frac{d}{dx} \left(x e^{x}\right) = \left(x + 1\right) e^{x}$$$.
Answer
$$$\frac{d}{dx} \left(x e^{x}\right) = \left(x + 1\right) e^{x}$$$A