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To differentiate the function \( y = 5x^2 + 4\tan x \) with respect to \( x \), we'll apply the basic rules of differentiation.

### Step-by-Step Differentiation:

1. **Differentiate \( 5x^2 \)**:
   \[
   \frac{d}{dx}(5x^2) = 5 \cdot \frac{d}{dx}(x^2) = 5 \cdot 2x = 10x
   \]

2. **Differentiate \( 4\tan x \)**:
   \[
   \frac{d}{dx}(4\tan x) = 4 \cdot \frac{d}{dx}(\tan x) = 4 \cdot \sec^2 x
   \]

3. **Combine the results**:
   \[
   \frac{dy}{dx} = 10x + 4\sec^2 x
   \]

### Final Answer:
\[
\frac{dy}{dx} = 10x + 4\sec^2 x
\]

Would you like more details or have any questions?

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Here are 5 related questions you might find helpful:

1. How do you differentiate \( \sec(x) \) or \( \csc(x) \)?
2. What is the derivative of a product of two functions?
3. How do you apply the chain rule in differentiation?
4. What are common derivatives involving trigonometric functions?
5. How can you find the second derivative of \( y = 5x^2 + 4\tan x \)?

**Tip:** Remember that when differentiating trigonometric functions, knowing the derivatives of basic functions like \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \) is crucial.

Differentiate y = 5x^2 + 4tan(x) with respect to x.

Learn how to differentiate y = 5x^2 + 4tan(x) with respect to x using the power rule and trigonometric derivatives. This detailed step-by-step solution is suitable for high school students in Grades 11-12.

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