To find the second derivative of the function $f(x) = (1 + 2x)^{10}$, we will follow these steps:

Step 1: First Derivative

Use the chain rule to differentiate $f(x)$ with respect to $x$:

$f'(x) = 10 \cdot (1 + 2x)^9 \cdot \frac{d}{dx}(1 + 2x)$

Since $\frac{d}{dx}(1 + 2x) = 2$, we get:

$f'(x) = 10 \cdot 2 \cdot (1 + 2x)^9 = 20(1 + 2x)^9$

Step 2: Second Derivative

Now, differentiate $f'(x) = 20(1 + 2x)^9$ again, applying the chain rule:

$f''(x) = 20 \cdot 9 \cdot (1 + 2x)^8 \cdot \frac{d}{dx}(1 + 2x)$

Again, $\frac{d}{dx}(1 + 2x) = 2$, so:

$f''(x) = 20 \cdot 9 \cdot 2 \cdot (1 + 2x)^8 = 360(1 + 2x)^8$

Final Answer:

The second derivative of $(1 + 2x)^{10}$ is:

$f''(x) = 360(1 + 2x)^8$

Would you like more details, or do you have any other questions?

Here are 5 related questions you can explore:

1. What is the third derivative of $(1 + 2x)^{10}$?
2. How does the chain rule apply to higher derivatives?
3. What is the general formula for the $n$-th derivative of $(1 + 2x)^{10}$?
4. How would you find the derivative of a more complex function like $(3 + 5x^2)^{10}$?
5. How do second derivatives help in finding concavity of a function?

Tip: The second derivative can be used to determine whether a function is concave up or concave down.