https://file.aiquickdraw.com/mathbot/file/1727840429552f5emnsyy.jpg

We are given the function \( g(x) = \frac{2x^2}{3x + 4} \) and need to find \( g'(1) \) using the quotient rule.

### Step 1: Recall the Quotient Rule
The quotient rule for derivatives states that if \( g(x) = \frac{f(x)}{h(x)} \), then
\[
g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{[h(x)]^2}
\]
where \( f(x) = 2x^2 \) (the numerator) and \( h(x) = 3x + 4 \) (the denominator).

### Step 2: Differentiate the Numerator and Denominator
- \( f(x) = 2x^2 \), so \( f'(x) = 4x \).
- \( h(x) = 3x + 4 \), so \( h'(x) = 3 \).

### Step 3: Apply the Quotient Rule
Now, applying the quotient rule:
\[
g'(x) = \frac{(4x)(3x + 4) - (2x^2)(3)}{(3x + 4)^2}
\]

### Step 4: Simplify and Substitute \( x = 1 \)
To find \( g'(1) \), substitute \( x = 1 \) into the expression:
- \( f'(1) = 4(1) = 4 \)
- \( f(1) = 2(1)^2 = 2 \)
- \( h(1) = 3(1) + 4 = 7 \)
- \( h'(1) = 3 \)

Substitute these values into the quotient rule:
\[
g'(1) = \frac{(4)(7) - (2)(3)}{7^2} = \frac{28 - 6}{49} = \frac{22}{49}
\]

Thus, \( g'(1) = \frac{22}{49} \).

Would you like further clarification or details?

---

Here are five related questions to expand understanding:
1. How does the quotient rule differ from the product rule in calculus?
2. What are some examples where the quotient rule is necessary in real-world applications?
3. How can you verify the result of a quotient rule derivative using another method?
4. What happens to the quotient rule when the denominator is a constant?
5. How can you extend the quotient rule to higher-order derivatives?

**Tip:** Always ensure the denominator function is non-zero at the point you're differentiating; otherwise, the function may not be defined.

Use the Quotient Rule to find g'(1) given that g(x) = (2x^2) / (3x + 4).

Learn how to apply the Quotient Rule to find the derivative of g(x) = (2x^2) / (3x + 4) and calculate g'(1) with a simplified step-by-step solution. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation