Questions: Consider the function f(x)=e^(3x^4-3x^2). To compute f'(x), we use the chain rule for f(x)=e^(g(x)), where g(x)=3x^4-3x^2 We express the derivative using the formula f'(x)=e^(g(x)) * g'(x)=g'(x) * e^(g(x)). The derivative is

Consider the function f(x)=e^(3x^4-3x^2). To compute f'(x), we use the chain rule for f(x)=e^(g(x)), where g(x)=3x^4-3x^2

We express the derivative using the formula f'(x)=e^(g(x)) * g'(x)=g'(x) * e^(g(x)). The derivative is

Transcript text: Consider the function $f(x)=e^{3 x^{4}-3 x^{2}}$. To compute $f^{\prime}(x)$, we use the chain rule for $f(x)=e^{g(x)}$, where $g(x)=3 x^{4}-3 x^{2}$ We express the derivative using the formula $f^{\prime}(x)=e^{g(x)} \cdot g^{\prime}(x)=g^{\prime}(x) \cdot e^{g(x)}$. The derivative is

Solution

Solution Steps

To find the derivative of the function $ f(x) = e^{3x^4 - 3x^2} $, we apply the chain rule. The chain rule states that if you have a composite function $ f(x) = e^{g(x)} $, then the derivative $ f'(x) $ is given by $ f'(x) = e^{g(x)} \cdot g'(x) $. Here, $ g(x) = 3x^4 - 3x^2 $. We first find $ g'(x) $ and then multiply it by $ e^{g(x)} $.

Step 1: Identify the Function and Its Components

We are given the function $ f(x) = e^{3x^4 - 3x^2} $. To find its derivative, we need to identify the inner function $ g(x) = 3x^4 - 3x^2 $.

Step 2: Differentiate the Inner Function

Calculate the derivative of the inner function $ g(x) $. The derivative is: \[ g'(x) = \frac{d}{dx}(3x^4 - 3x^2) = 12x^3 - 6x \]

Step 3: Apply the Chain Rule

Using the chain rule for derivatives, the derivative of $ f(x) = e^{g(x)} $ is: \[ f'(x) = e^{g(x)} \cdot g'(x) \] Substituting the expressions for $ g(x) $ and $ g'(x) $, we have: \[ f'(x) = e^{3x^4 - 3x^2} \cdot (12x^3 - 6x) \]