Find the derivative of 2x^2+3 over 2x+4 using the definition of derivative

To find the derivative of \( f(x) = \frac{2x^2 + 3}{2x + 4} \) using the definition of the derivative, we'll follow the formal limit definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Here are the steps to calculate this derivative:

1. **Substitute \( x+h \) into the function:**
   First, compute \( f(x+h) \):
   \[
   f(x+h) = \frac{2(x+h)^2 + 3}{2(x+h) + 4}
   \]

2. **Simplify \( f(x+h) - f(x) \):**
   Calculate the difference \( f(x+h) - f(x) \).

3. **Divide by \( h \):**
   Express the difference quotient as \( \frac{f(x+h) - f(x)}{h} \).

4. **Take the limit as \( h \to 0 \):**
   Evaluate the limit of the difference quotient to obtain the derivative.

Let's go through the steps to find the derivative.

### Step 1: Calculate \( f(x+h) \)
\[
f(x+h) = \frac{2(x+h)^2 + 3}{2(x+h) + 4}
\]
Expanding this:
\[
f(x+h) = \frac{2(x^2 + 2xh + h^2) + 3}{2x + 2h + 4}
       = \frac{2x^2 + 4xh + 2h^2 + 3}{2x + 2h + 4}
\]

### Step 2: Compute \( f(x+h) - f(x) \)
\[
f(x+h) - f(x) = \frac{2x^2 + 4xh + 2h^2 + 3}{2x + 2h + 4} - \frac{2x^2 + 3}{2x + 4}
\]
To subtract these two fractions, we'll need a common denominator.

The common denominator is \( (2x + 2h + 4)(2x + 4) \). Let's compute the difference of numerators, simplifying the expression.

### Step 3 and Step 4: Simplification and Limit Evaluation
At this point, the algebra becomes more involved. I will proceed with the detailed calculation and take the limit as \( h \to 0 \).The derivative of \( f(x) = \frac{2x^2 + 3}{2x + 4} \) using the definition of the derivative is:

\[
f'(x) = \frac{x^2 + 4x - \frac{3}{2}}{x^2 + 4x + 4}
\]

Would you like further details on any of the steps, or have any questions?

Here are five related questions to expand your understanding:
1. How would the derivative change if the function was \( \frac{x^2 + 3x}{2x + 4} \)?
2. What is the interpretation of the derivative in terms of the slope of the tangent line?
3. How do you find the second derivative of \( \frac{2x^2 + 3}{2x + 4} \)?
4. How can you apply the quotient rule to differentiate this function without using the definition of the derivative?
5. What happens to the derivative at points where the denominator equals zero?

**Tip:** When calculating limits involving fractions, simplifying the numerator and denominator separately often helps to avoid unnecessary complexity.

Learn how to calculate the derivative of the function (2x^2 + 3) / (2x + 4) using the definition of derivative. This problem involves applying the limit definition of derivative step-by-step, including substitution, simplification, and limit evaluation. Suitable for advanced high school or college students studying calculus.

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