(3x⁴+4x²+6x)/(2x) differentiate this expression

To differentiate the given expression:

\[
\frac{3x^4 + 4x^2 + 6x}{2x}
\]

we can first simplify the expression by dividing each term in the numerator by \(2x\):

\[
\frac{3x^4}{2x} + \frac{4x^2}{2x} + \frac{6x}{2x}
= \frac{3}{2}x^3 + 2x + 3
\]

Now, we differentiate term by term:

1. For \(\frac{3}{2}x^3\), use the power rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\):
   \[
   \frac{d}{dx}\left(\frac{3}{2}x^3\right) = \frac{9}{2}x^2
   \]

2. For \(2x\), the derivative is:
   \[
   \frac{d}{dx}(2x) = 2
   \]

3. For the constant \(3\), the derivative is:
   \[
   \frac{d}{dx}(3) = 0
   \]

So, the derivative of the simplified expression is:

\[
\frac{9}{2}x^2 + 2
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you apply the power rule to differentiate polynomials?
2. How can you differentiate rational expressions in general?
3. What is the chain rule and when is it necessary in differentiation?
4. How do constants affect the differentiation process?
5. How do you verify if your differentiation is correct?

### Tip:
Always simplify expressions before differentiating to make the process easier!

Learn how to differentiate the rational expression (3x⁴ + 4x² + 6x) / (2x). This step-by-step solution covers polynomial simplification and the use of the power rule. Suitable for advanced high school students in Grades 11-12.

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