find the derivative of the function f(x) = (3x^2 - 2x + 5)^10

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!

Learn how to find the derivative of the function f(x) = (3x^2 - 2x + 5)^10 using the chain rule. This detailed solution covers key calculus concepts such as the power rule and the chain rule. Suitable for advanced high school or early college students.

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