Questions: Find the derivative of the function. y = -3 e^(-5 x^2) dy/dx = □

Transcript text: Find the derivative of the function. \[ \begin{array}{l} y=-3 e^{-5 x^{2}} \\ \frac{d y}{d x}=\square \end{array} \]

Solution

Solution Steps

To find the derivative of the function \( y = -3 e^{-5 x^2} \), we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is \( -3 e^u \) where \( u = -5 x^2 \), and the inner function is \( -5 x^2 \).

Step 1: Define the Function

We start with the function given in the problem: \[ y = -3 e^{-5 x^2} \]

Step 2: Apply the Chain Rule

To find the derivative \(\frac{dy}{dx}\), we apply the chain rule. The outer function is \(-3 e^u\) where \(u = -5 x^2\). The derivative of the outer function is: \[ \frac{d}{du}(-3 e^u) = -3 e^u \] The inner function \(u = -5 x^2\) has a derivative: \[ \frac{du}{dx} = -10 x \]

Step 3: Combine Derivatives

Using the chain rule, we combine the derivatives: \[ \frac{dy}{dx} = \frac{d}{du}(-3 e^u) \cdot \frac{du}{dx} = -3 e^{-5 x^2} \cdot (-10 x) = 30 x e^{-5 x^2} \]