differentiate the following function. f(x)=e^(-4x^2 - 6x)

Explanation:

Step1: Identify the outer - inner functions

Let $u = - 4x^{2}-6x$, then $y = e^{u}$.

Step2: Differentiate the outer function

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$.

Step3: Differentiate the inner function

The derivative of $u=-4x^{2}-6x$ with respect to $x$ is $\frac{du}{dx}=-8x - 6$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=-8x - 6$ and $u=-4x^{2}-6x$ back in, we get $\frac{dy}{dx}=e^{-4x^{2}-6x}(-8x - 6)$.

Answer:

$(-8x - 6)e^{-4x^{2}-6x}$