To find the derivative of the function $f(x) = (3x^2 - 2x + 5)^{10}$, we will apply the chain rule.

The chain rule states that if you have a composite function $f(g(x))$, then its derivative is:

$f'(x) = f'(g(x)) \cdot g'(x)$

Here, $f(x) = (u(x))^{10}$, where $u(x) = 3x^2 - 2x + 5$.

Step 1: Differentiate the outer function

The derivative of $(u(x))^{10}$ with respect to $u(x)$ is:

$\frac{d}{du} \left[ u(x)^{10} \right] = 10 \cdot (u(x))^9 = 10 \cdot (3x^2 - 2x + 5)^9$

Step 2: Differentiate the inner function

Now, find the derivative of $u(x) = 3x^2 - 2x + 5$:

$u'(x) = \frac{d}{dx} \left[ 3x^2 - 2x + 5 \right] = 6x - 2$

Step 3: Apply the chain rule

Now multiply the derivative of the outer function by the derivative of the inner function:

$f'(x) = 10 \cdot (3x^2 - 2x + 5)^9 \cdot (6x - 2)$

Thus, the derivative of the function is:

$f'(x) = 10(3x^2 - 2x + 5)^9(6x - 2)$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the second derivative of $f(x) = (3x^2 - 2x + 5)^{10}$?
2. How would you find the derivative of $f(x) = (2x^3 - x + 4)^7$?
3. What is the derivative of $f(x) = (x^2 + 1)^5$?
4. Can you find the derivative of $f(x) = \sin^4(x^2 - 3x)$?
5. How do you apply the chain rule to $f(x) = \ln(5x^2 - 3x + 2)$?

Tip: Always check if the chain rule applies when differentiating composite functions!