If $y = 5e^{2x} + 4e^{3x}$, then $\frac{d^2y}{dx^2}$ equals:

Given $y = 5e^{2x} + 4e^{3x}$

When differentiating $e^{ax}$, the derivative is $a \cdot e^{ax}$

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Finding the first derivative:

For $5e^{2x}$: the derivative is $5 \times 2 \times e^{2x} = 10e^{2x}$

For $4e^{3x}$: the derivative is $4 \times 3 \times e^{3x} = 12e^{3x}$

$\frac{dy}{dx} = 10e^{2x} + 12e^{3x}$

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Finding the second derivative:

For $10e^{2x}$: the derivative is $10 \times 2 \times e^{2x} = 20e^{2x}$

For $12e^{3x}$: the derivative is $12 \times 3 \times e^{3x} = 36e^{3x}$

$\frac{d^2y}{dx^2} = 20e^{2x} + 36e^{3x}$

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Factoring the result:

$\frac{d^2y}{dx^2} = 20e^{2x} + 36e^{3x}$

$\frac{d^2y}{dx^2} = 4(5e^{2x} + 9e^{3x})$

Therefore, $\frac{d^2y}{dx^2} = 4(5e^{2x} + 9e^{3x})$