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Second derivative of $$$2 e^{x}$$$
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Find $$$\frac{d^{2}}{dx^{2}} \left(2 e^{x}\right)$$$.
Solution
Find the first derivative $$$\frac{d}{dx} \left(2 e^{x}\right)$$$
Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = e^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 e^{x}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(e^{x}\right)\right)}$$The derivative of the exponential is $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} = 2 {\color{red}\left(e^{x}\right)}$$Thus, $$$\frac{d}{dx} \left(2 e^{x}\right) = 2 e^{x}$$$.
Next, $$$\frac{d^{2}}{dx^{2}} \left(2 e^{x}\right) = \frac{d}{dx} \left(2 e^{x}\right)$$$
Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = e^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 e^{x}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(e^{x}\right)\right)}$$The derivative of the exponential is $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} = 2 {\color{red}\left(e^{x}\right)}$$Thus, $$$\frac{d}{dx} \left(2 e^{x}\right) = 2 e^{x}$$$.
Therefore, $$$\frac{d^{2}}{dx^{2}} \left(2 e^{x}\right) = 2 e^{x}$$$.
Answer
$$$\frac{d^{2}}{dx^{2}} \left(2 e^{x}\right) = 2 e^{x}$$$A