Question

hw_3.1_the chain rule
due sunday by 11:59pm points 100.75 submitting
hw_3.1_the chain rule

1. submit answer practice similar

attempt 1: 10 attempts remaining.
find the derivative of the function
y = e^{2x^{2}+6x + 2}
using the chain rule for exponential functions.
$\frac{dy}{dx}=$

Explanation:

Step1: Identify outer - inner functions

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

Step2: Differentiate outer function

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

Step3: Differentiate inner function

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

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=4x + 6$ back in, and replace $u$ with $2x^{2}+6x + 2$. So $\frac{dy}{dx}=e^{2x^{2}+6x + 2}(4x + 6)$.

Answer:

$(4x + 6)e^{2x^{2}+6x + 2}$