Questions: Find the derivative, f'(x), of the function f(x)=x^3 e^x A. f'(x)=x^3 e^x+x^2 e^x B. f'(x)=x^3 θ^x+3 x^2 θ^x C. f'(x)=x^3 0^x+3 x^2 D. f'(x)=x^3 e^x+3 e^x E. f'(x)=x^2 0^x+3 x θ^x

Find the derivative, f'(x), of the function f(x)=x^3 e^x A. f'(x)=x^3 e^x+x^2 e^x B. f'(x)=x^3 θ^x+3 x^2 θ^x C. f'(x)=x^3 0^x+3 x^2 D. f'(x)=x^3 e^x+3 e^x E. f'(x)=x^2 0^x+3 x θ^x
Transcript text: Find the derivative, $f^{\prime}(x)$, of the function $f(x)=x^{3} e^{x}$ A. $f^{\prime}(x)=x^{3} e^{x}+x^{2} e^{x}$ B. $f^{\prime}(x)=x^{3} \theta^{x}+3 x^{2} \theta^{x}$ C. $f^{\prime}(x)=x^{3} 0^{x}+3 x^{2}$ D. $f^{\prime}(x)=x^{3} e^{x}+3 e^{x}$ E. $f^{\prime}(x)=x^{2} 0^{x}+3 x \theta^{x}$
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Solution

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Solution Steps

Step 1: Define the Function

We start with the function \( f(x) = x^3 e^x \).

Step 2: Apply the Product Rule

To find the derivative \( f^{\prime}(x) \), we apply the product rule. Let \( u(x) = x^3 \) and \( v(x) = e^x \). The derivatives are:

  • \( u^{\prime}(x) = 3x^2 \)
  • \( v^{\prime}(x) = e^x \)

According to the product rule: \[ f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x) = 3x^2 e^x + x^3 e^x \]

Step 3: Simplify the Derivative

We can factor out \( e^x \) from the expression: \[ f^{\prime}(x) = e^x (x^3 + 3x^2) \]

Final Answer

The derivative of the function \( f(x) = x^3 e^x \) is: \[ \boxed{f^{\prime}(x) = e^x (x^3 + 3x^2)} \]

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