solve derrivative of this equation x^7+x^3+2x

To find the derivative of the function \( f(x) = x^7 + x^3 + 2x \), we'll apply the basic rules of differentiation.

### Function:
\[
f(x) = x^7 + x^3 + 2x
\]

### Derivative:
The derivative of each term is found using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

1. For \( x^7 \):
   \[
   \frac{d}{dx}(x^7) = 7x^6
   \]
   
2. For \( x^3 \):
   \[
   \frac{d}{dx}(x^3) = 3x^2
   \]

3. For \( 2x \):
   \[
   \frac{d}{dx}(2x) = 2
   \]

### Combined Derivative:
\[
f'(x) = 7x^6 + 3x^2 + 2
\]

So, the derivative of \( f(x) = x^7 + x^3 + 2x \) is:

\[
f'(x) = 7x^6 + 3x^2 + 2
\]

Would you like more details or have any questions? Here are some related questions:

1. How do you find the derivative of a product of functions?
2. What is the significance of the derivative in understanding the behavior of a function?
3. How do you differentiate a function involving exponential terms?
4. What are the applications of derivatives in real-world problems?
5. How can you find the second derivative, and what does it represent?

**Tip:** The derivative of a function gives the slope of the tangent line to the curve at any given point, providing insights into the function's rate of change.

Learn how to find the derivative of the function f(x) = x^7 + x^3 + 2x using the power rule and basic differentiation principles. This problem involves differentiation of polynomial functions and is suitable for advanced high school students or college-level calculus courses.

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