Questions: Differentiate the following function. f(x)=x^4 e^4 x f'(x)=

Transcript text: Differentiate the following function. \[ \begin{array}{l} f(x)=x^{4} e^{4 x} \\ f^{\prime}(x)=\square \end{array} \] $\square$

Solution

Solution Steps

To differentiate the function $ f(x) = x^4 e^{4x} $, we will use the product rule. The product rule states that if you have a function that is the product of two functions, $ u(x) $ and $ v(x) $, then the derivative is given by $ u'(x)v(x) + u(x)v'(x) $. Here, let $ u(x) = x^4 $ and $ v(x) = e^{4x} $. We will find the derivatives $ u'(x) $ and $ v'(x) $ and then apply the product rule.

Step 1: Define the Function

We start with the function defined as: \[ f(x) = x^4 e^{4x} \]

Step 2: Apply the Product Rule

To differentiate $ f(x) $, we apply the product rule. Let: \[ u(x) = x^4 \quad \text{and} \quad v(x) = e^{4x} \] Then, the derivatives are: \[ u'(x) = 4x^3 \quad \text{and} \quad v'(x) = 4e^{4x} \]

Step 3: Compute the Derivative

Using the product rule: \[ f'(x) = u'(x)v(x) + u(x)v'(x) \] Substituting the values we found: \[ f'(x) = (4x^3)(e^{4x}) + (x^4)(4e^{4x}) \] This simplifies to: \[ f'(x) = 4x^3 e^{4x} + 4x^4 e^{4x} \] Factoring out the common terms gives: \[ f'(x) = 4e^{4x}(x^3 + x^4) \]