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The given function is:

\[
f(x) = 2x^5 + 7
\]

To find the derivative of this function, \( f'(x) \), we apply the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \), and the derivative of a constant is 0.

### Step-by-step solution:

1. For the term \( 2x^5 \):
   - The derivative of \( x^5 \) is \( 5x^4 \).
   - Multiply this by the constant 2: 
   \[
   \frac{d}{dx} [2x^5] = 2 \cdot 5x^4 = 10x^4.
   \]

2. For the constant \( 7 \):
   - The derivative of any constant is 0:
   \[
   \frac{d}{dx} [7] = 0.
   \]

### Therefore, the derivative is:
\[
f'(x) = 10x^4.
\]

Would you like further explanation on differentiation or any other questions?

Here are 5 related questions to expand the topic:
1. How do you apply the chain rule in differentiation?
2. What is the derivative of \( f(x) = 3x^4 + 6x^2 + 9 \)?
3. Can you explain the product rule for differentiation?
4. How do you differentiate trigonometric functions like \( \sin(x) \) or \( \cos(x) \)?
5. What is the second derivative of \( f(x) = 2x^5 + 7 \)?

**Tip:** Always simplify your function as much as possible before applying differentiation rules.

Learn how to differentiate the function f(x) = 2x^5 + 7. The problem demonstrates the application of the power rule in calculus to find the derivative. A step-by-step solution is provided, suitable for students in Grades 10-12 or first-year university calculus.

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