If g = f', determine:

d/dx 6x^2 * e ^ (x * f(2x))

To find the derivative of \( 6x^2 \cdot e^{x \cdot f(2x)} \), we will apply the product rule and the chain rule.

Let’s break it down step by step:

### Step 1: Apply the Product Rule
The product rule states:
\[
\frac{d}{dx} (u \cdot v) = u'v + uv'
\]
Here, let:
- \( u = 6x^2 \)
- \( v = e^{x \cdot f(2x)} \)

We need to find \( u' \) and \( v' \).

### Step 2: Find \( u' \)
The derivative of \( u = 6x^2 \) is straightforward:
\[
u' = \frac{d}{dx} 6x^2 = 12x
\]

### Step 3: Find \( v' \)
Now we need to differentiate \( v = e^{x \cdot f(2x)} \). This involves applying the chain rule:
\[
v' = \frac{d}{dx} e^{x \cdot f(2x)} = e^{x \cdot f(2x)} \cdot \frac{d}{dx} (x \cdot f(2x))
\]

To differentiate \( x \cdot f(2x) \), use the product rule again:
\[
\frac{d}{dx} (x \cdot f(2x)) = \frac{d}{dx} x \cdot f(2x) + x \cdot \frac{d}{dx} f(2x)
\]

- The derivative of \( x \) is 1.
- The derivative of \( f(2x) \) using the chain rule is \( f'(2x) \cdot 2 = g(2x) \cdot 2 \) (since \( g(x) = f'(x) \)).

So, we have:
\[
\frac{d}{dx} (x \cdot f(2x)) = f(2x) + 2x \cdot g(2x)
\]

Thus:
\[
v' = e^{x \cdot f(2x)} \cdot (f(2x) + 2x \cdot g(2x))
\]

### Step 4: Combine using the Product Rule
Now, using the product rule:
\[
\frac{d}{dx} \left( 6x^2 \cdot e^{x \cdot f(2x)} \right) = u'v + uv'
\]

Substituting the values we found for \( u' \), \( v \), and \( v' \):
\[
= 12x \cdot e^{x \cdot f(2x)} + 6x^2 \cdot e^{x \cdot f(2x)} \cdot (f(2x) + 2x \cdot g(2x))
\]

### Final Answer:
\[
\frac{d}{dx} \left( 6x^2 \cdot e^{x \cdot f(2x)} \right) = 12x \cdot e^{x \cdot f(2x)} + 6x^2 \cdot e^{x \cdot f(2x)} \cdot (f(2x) + 2x \cdot g(2x))
\]

Let me know if you'd like to see more details or have any questions!

### Related Questions:
1. How is the product rule used in differentiation?
2. Can you explain how the chain rule applies to exponential functions?
3. What is the significance of \( f'(x) = g(x) \) in this problem?
4. How do you differentiate a composite function involving another function?
5. How would the derivative change if \( f(x) \) was a specific function, like \( \sin(x) \)?

### Tip:
Always check for opportunities to apply both the product and chain rules in combination, as in this problem!

If g = f', determine: d/dx 6x^2 * e^(x * f(2x))

Discover how to differentiate the expression 6x^2 * e^(x * f(2x)) using both the product and chain rules. This detailed solution covers key calculus concepts such as composite functions and the use of derivatives of exponential functions, suitable for undergraduate students.

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