what is the second derivative of (1+2x)tot de macht 10

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.

Discover how to calculate the second derivative of the function (1 + 2x)^10 using the chain rule. This step-by-step solution includes the application of basic differentiation and higher-order derivatives. Suitable for students in Grades 10-12 studying calculus.

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