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.