Question

attempt 1: 10 attempts remaining. find the derivative of the function ( y = e^{6x^2 + 2x + 3} ) using the chain rule for exponential functions. ( \frac{dy}{dx} = ) input box submit answer next item

Explanation:

Step1: Identify the outer and inner functions

The function is \( y = e^{6x^2 + 2x + 3} \). The outer function is \( f(u)=e^u \) where \( u = 6x^2 + 2x + 3 \) (the inner function).

Step2: Differentiate the outer function

The derivative of \( e^u \) with respect to \( u \) is \( f'(u)=e^u \).

Step3: Differentiate the inner function

Differentiate \( u = 6x^2 + 2x + 3 \) with respect to \( x \). Using the power rule, \( \frac{du}{dx}=12x + 2 \).

Step4: Apply the chain rule

The chain rule states that \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). Substituting the derivatives from Step2 and Step3, we get:
\( \frac{dy}{dx}=e^{6x^2 + 2x + 3}\cdot(12x + 2) \)

Answer:

\( (12x + 2)e^{6x^2 + 2x + 3} \)