To find $f'(3)$, we first need to determine the derivative $f'(x)$ of the function $f(x) = 4x^2 - 2e^x$, and then evaluate it at $x = 3$.

Step 1: Differentiate $f(x)$

The derivative of $f(x) = 4x^2 - 2e^x$ is: $f'(x) = \frac{d}{dx}(4x^2) - \frac{d}{dx}(2e^x)$

1. The derivative of $4x^2$ is: $\frac{d}{dx}(4x^2) = 8x$

2. The derivative of $-2e^x$ is: $\frac{d}{dx}(-2e^x) = -2e^x$

Thus: $f'(x) = 8x - 2e^x$

Step 2: Evaluate $f'(x)$ at $x = 3$

Substitute $x = 3$ into $f'(x)$: $f'(3) = 8(3) - 2e^3$ $f'(3) = 24 - 2e^3$

Final Answer:

$f'(3) = 24 - 2e^3$

This is the exact value. If needed, you can calculate the approximate numerical value of $e^3$.

Would you like me to calculate the approximate numerical value? Let me know if you have any questions!

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Related Questions:

1. How do you compute derivatives of exponential functions in general?
2. What is the meaning of $f'(x)$ in the context of a function?
3. How would $f'(x)$ change if $f(x)$ included $e^{-x}$?
4. Can we graph $f(x)$ and $f'(x)$ to visualize the behavior?
5. What are common applications of exponential functions in real-world problems?

Tip:

Always remember that $e^x$ is its own derivative, which makes exponential functions unique and widely applicable in calculus!