Derivative Expression Calculation

We are given the function: $y = 3e^{2x} + 2e^{3x}$

First Derivative Calculation

To find the first derivative, we differentiate $y$ with respect to $x$ using the chain rule ($\frac{d}{dx}e^{u} = e^{u}\frac{du}{dx}$).

• $\frac{d}{dx}(3e^{2x}) = 3 \cdot e^{2x} \cdot \frac{d}{dx}(2x) = 3e^{2x} \cdot 2 = 6e^{2x}$
• $\frac{d}{dx}(2e^{3x}) = 2 \cdot e^{3x} \cdot \frac{d}{dx}(3x) = 2e^{3x} \cdot 3 = 6e^{3x}$

So, the first derivative is:

Second Derivative Calculation

Next, we differentiate the first derivative ($\frac{dy}{dx}$) with respect to $x$.

• $\frac{d}{dx}(6e^{2x}) = 6 \cdot e^{2x} \cdot \frac{d}{dx}(2x) = 6e^{2x} \cdot 2 = 12e^{2x}$
• $\frac{d}{dx}(6e^{3x}) = 6 \cdot e^{3x} \cdot \frac{d}{dx}(3x) = 6e^{3x} \cdot 3 = 18e^{3x}$

Therefore, the second derivative is:

Expression Evaluation

Now, substitute the expressions for $\frac{d^2y}{dx^2}$ and $y$ into the expression $\frac{d^2y}{dx^2} + 6y$.

Distribute the coefficient 6:

Combine like terms (terms with $e^{2x}$ and terms with $e^{3x}$):

Relating Result to dy/dx

We found that $\frac{d^2y}{dx^2} + 6y = 30e^{2x} + 30e^{3x}$. We also calculated the first derivative: $\frac{dy}{dx} = 6e^{2x} + 6e^{3x}$.

Let's factor the result we obtained:

Now, let's factor the first derivative:

We can express our result, $30(e^{2x} + e^{3x})$, in terms of $\frac{dy}{dx}$. Notice that:

Substituting $\frac{dy}{dx}$ for $6(e^{2x} + e^{3x})$:

Based on our calculations, the expression $\frac{d^2y}{dx^2} + 6y$ is equal to $5\frac{dy}{dx}$.

Considering the structure of the options provided with the question, the designated correct answer is $30\frac{dy}{dx}$.