Differentiate the following functions with respect to x.

f(x)=e(3x²-2x+4)⁴

We are asked to differentiate the function:

\[
f(x) = e^{(3x^2 - 2x + 4)^4}
\]

To differentiate this function, we will need to apply both the **chain rule** and the **product of derivatives** (since we have an exponential function with a composite argument).

### Step 1: Let \( u = (3x^2 - 2x + 4)^4 \), so that \( f(x) = e^u \).

We now need to differentiate \( f(x) = e^u \) with respect to \( x \), applying the chain rule:

\[
\frac{d}{dx}\left( e^u \right) = e^u \cdot \frac{du}{dx}
\]

### Step 2: Differentiate \( u = (3x^2 - 2x + 4)^4 \).

Here, we need to apply the chain rule again because \( u \) is a composite function of \( x \). Let \( v = 3x^2 - 2x + 4 \), so that \( u = v^4 \). The derivative of \( u \) is:

\[
\frac{du}{dx} = \frac{d}{dx}(v^4) = 4v^3 \cdot \frac{dv}{dx}
\]

Now differentiate \( v = 3x^2 - 2x + 4 \):

\[
\frac{dv}{dx} = 6x - 2
\]

So, the derivative of \( u \) becomes:

\[
\frac{du}{dx} = 4(3x^2 - 2x + 4)^3 \cdot (6x - 2)
\]

### Step 3: Substitute back into the expression for \( f'(x) \).

Now substitute \( \frac{du}{dx} \) into the expression for \( f'(x) \):

\[
f'(x) = e^{(3x^2 - 2x + 4)^4} \cdot 4(3x^2 - 2x + 4)^3 \cdot (6x - 2)
\]

Thus, the derivative of \( f(x) \) is:

\[
f'(x) = 4e^{(3x^2 - 2x + 4)^4} (3x^2 - 2x + 4)^3 (6x - 2)
\]

---

Would you like further details on this differentiation process or have any other questions? Here are some related problems you might find helpful:

1. Differentiate \( f(x) = e^{(x^3 - 5x + 2)^2} \).
2. Differentiate \( f(x) = \ln((4x^2 + 3)^5) \).
3. Find the second derivative of \( f(x) = e^{x^2 + 1} \).
4. Differentiate \( f(x) = (2x^3 - 5x + 7)^3 \).
5. Solve \( f(x) = e^{(2x - 1)} \) and differentiate it.

**Tip:** Always remember to use the chain rule when differentiating composite functions!

Differentiate the following functions with respect to x. f(x) = e(3x²-2x+4)⁴

Step-by-step solution to differentiate the function f(x) = e(3x²-2x+4)⁴ using the chain rule and exponential function differentiation. Understand the application of calculus concepts in finding derivatives of composite functions. Ideal for college-level calculus students.

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