Differentiate 8e-x+2ex w.r.t x

The correct option is 2e^x-8e^-x.

To find the derivative of the given function with respect to $x$, let us define the function as:

$$y=8e^{-x}+2e^x$$

We need to find the derivative $\frac{dy}{dx}$. According to the sum rule of differentiation, we can differentiate each term individually:

$$\frac{dy}{dx}=\frac{d}{dx}\left(8e^{-x}\right)+\frac{d}{dx}\left(2e^x\right)$$

Now, let's apply the standard rules of differentiation:

1. The constant multiple rule: $\frac{d}{dx}[c\cdot f(x\left)\right]=c\cdot \frac{d}{dx}\left[f\right(x\left)\right]$

2. The derivative of $e^x$ with respect to $x$ is $e^x$.

3. The derivative of $e^{-x}$ with respect to $x$, using the chain rule, is $-e^{-x}$.

Applying these rules, we get:

$$\frac{dy}{dx}=8\cdot (-e^{-x})+2\cdot \left(e^x\right)$$

Simplifying the expression:

$$\frac{dy}{dx}=-8e^{-x}+2e^x$$

Rearranging the terms to match the correct option format:

$$\frac{dy}{dx}=2e^x-8e^{-x}$$