Questions: Find the derivative of the function. y=-4 e^(2 x^2)

Solution Steps

To find the derivative of the function \( y = -4 e^{2x^2} \), we will use the chain rule. The chain rule is used when differentiating a composite function. Here, the outer function is the exponential function and the inner function is \( 2x^2 \). We will differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function.

We start with the function given by \[ y = -4 e^{2x^2}. \]

To find the derivative \( \frac{dy}{dx} \), we apply the chain rule. The outer function is \( -4 e^u \) where \( u = 2x^2 \). The derivative of the outer function is \[ -4 e^u, \] and the derivative of the inner function \( u = 2x^2 \) is \[ \frac{du}{dx} = 4x. \]

Using the chain rule, we combine the derivatives: \[ \frac{dy}{dx} = -4 e^{2x^2} \cdot 4x = -16x e^{2x^2}. \]

Final Answer