Questions: f(x)=(x^2+1)e^(4x)

Solution Steps

To find the derivative of the function \( f(x) = (x^2 + 1) e^{4x} \), we will use the product rule of differentiation. The product rule states that if you have a function \( f(x) = u(x) \cdot v(x) \), then the derivative \( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \). Here, \( u(x) = x^2 + 1 \) and \( v(x) = e^{4x} \). We will differentiate each part separately and then apply the product rule.

We start with the function defined as: \[ f(x) = (x^2 + 1) e^{4x} \]

We identify the components of the function:

• Let \( u = x^2 + 1 \)
• Let \( v = e^{4x} \)

Next, we compute the derivatives of \( u \) and \( v \): \[ u' = \frac{d}{dx}(x^2 + 1) = 2x \] \[ v' = \frac{d}{dx}(e^{4x}) = 4e^{4x} \]

Using the product rule, we find the derivative \( f'(x) \): \[ f'(x) = u'v + uv' = (2x)e^{4x} + (x^2 + 1)(4e^{4x}) \]

We can factor out \( e^{4x} \) from the expression: \[ f'(x) = e^{4x}(2x + 4(x^2 + 1)) = e^{4x}(2x + 4x^2 + 4) \]

Final Answer