https://file.aiquickdraw.com/mathbot/file/1722404700978ikqgnow9.pdf

To solve problem 3, we need to find the derivative of the function \( f(x) = 5x^2 + 11 \) using the limit definition of a derivative. The limit definition of a derivative is given by:

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

Let's follow the steps to find \( f'(x) \).

1. **Function Definition**: Start with \( f(x) = 5x^2 + 11 \).

2. **Increment Function**: Find \( f(x+h) \).
   \[ f(x+h) = 5(x+h)^2 + 11 \]
   \[ f(x+h) = 5(x^2 + 2xh + h^2) + 11 \]
   \[ f(x+h) = 5x^2 + 10xh + 5h^2 + 11 \]

3. **Difference Function**: Calculate \( f(x+h) - f(x) \).
   \[ f(x+h) - f(x) = (5x^2 + 10xh + 5h^2 + 11) - (5x^2 + 11) \]
   \[ f(x+h) - f(x) = 10xh + 5h^2 \]

4. **Quotient**: Form the difference quotient.
   \[ \frac{f(x+h) - f(x)}{h} = \frac{10xh + 5h^2}{h} \]
   \[ \frac{f(x+h) - f(x)}{h} = 10x + 5h \]

5. **Limit**: Take the limit as \( h \) approaches 0.
   \[ f'(x) = \lim_{h \to 0} (10x + 5h) \]
   \[ f'(x) = 10x \]

So, the derivative of \( f(x) = 5x^2 + 11 \) is:

\[ f'(x) = 10x \]

### Steps Summarized:
1. Substitute \( x+h \) into the function: \( f(x+h) = 5(x+h)^2 + 11 = 5x^2 + 10xh + 5h^2 + 11 \)
2. Subtract \( f(x) \) from \( f(x+h) \): \( f(x+h) - f(x) = 10xh + 5h^2 \)
3. Divide by \( h \): \( \frac{10xh + 5h^2}{h} = 10x + 5h \)
4. Take the limit as \( h \to 0 \): \( \lim_{h \to 0} (10x + 5h) = 10x \)

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the derivative of a polynomial function using the power rule?
2. What is the limit definition of a derivative?
3. Can you explain the product rule for differentiation?
4. How do you differentiate a function using the chain rule?
5. What are the basic rules for differentiation?
6. How can you find the second derivative of a function?
7. What is the geometric interpretation of a derivative?
8. How do you differentiate trigonometric functions?

**Tip:** When using the limit definition of a derivative, always expand and simplify the function before taking the limit. This makes it easier to isolate and cancel out the \( h \) terms.

Learn how to calculate the derivative of the polynomial function 5x^2 + 11 using the limit definition of a derivative. Follow a detailed step-by-step explanation with formulas and examples. Suitable for high school students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation