Given  
f
(
x
)
=
√
x
−
6
 , find  
f
′
(
7
)
  using the definition of a derivative.

To find the derivative of the function \( f(x) = \sqrt{x} - 6 \) at \( x = 7 \) using the definition of a derivative, we need to apply the limit definition of a derivative:

\[
f'(7) = \lim_{h \to 0} \frac{f(7+h) - f(7)}{h}
\]

Let's break this down step by step:

1. **Evaluate \( f(7) \)**:
   \[
   f(7) = \sqrt{7} - 6
   \]

2. **Evaluate \( f(7 + h) \)**:
   \[
   f(7 + h) = \sqrt{7 + h} - 6
   \]

3. **Set up the difference quotient**:
   \[
   \frac{f(7 + h) - f(7)}{h} = \frac{\left( \sqrt{7 + h} - 6 \right) - \left( \sqrt{7} - 6 \right)}{h}
   \]
   Simplify the numerator:
   \[
   = \frac{\sqrt{7 + h} - \sqrt{7}}{h}
   \]

4. **Multiply numerator and denominator by the conjugate**:
   To simplify this expression, we multiply both the numerator and denominator by the conjugate of the numerator:
   \[
   \frac{\sqrt{7 + h} - \sqrt{7}}{h} \cdot \frac{\sqrt{7 + h} + \sqrt{7}}{\sqrt{7 + h} + \sqrt{7}} = \frac{\left( \sqrt{7 + h} - \sqrt{7} \right) \left( \sqrt{7 + h} + \sqrt{7} \right)}{h \left( \sqrt{7 + h} + \sqrt{7} \right)}
   \]
   Simplify the numerator using the difference of squares:
   \[
   = \frac{(7 + h) - 7}{h \left( \sqrt{7 + h} + \sqrt{7} \right)} = \frac{h}{h \left( \sqrt{7 + h} + \sqrt{7} \right)}
   \]

5. **Simplify and take the limit**:
   Cancel out \( h \) in the numerator and denominator:
   \[
   = \frac{1}{\sqrt{7 + h} + \sqrt{7}}
   \]
   Now, take the limit as \( h \to 0 \):
   \[
   \lim_{h \to 0} \frac{1}{\sqrt{7 + h} + \sqrt{7}} = \frac{1}{\sqrt{7} + \sqrt{7}} = \frac{1}{2\sqrt{7}}
   \]

Therefore, the derivative of \( f(x) = \sqrt{x} - 6 \) at \( x = 7 \) is:
\[
f'(7) = \frac{1}{2\sqrt{7}}
\]

Would you like to go deeper into any step or have any questions?

---

Here are 5 related questions for further practice:
1. What is the derivative of \( f(x) = \sqrt{x} - 6 \) at \( x = 9 \)?
2. How would you find the derivative of \( f(x) = \sqrt{x} \) at a general point \( x = a \)?
3. Can you apply the same method to find the derivative of \( f(x) = x^2 - 5x \)?
4. How can you use the chain rule to differentiate \( f(x) = \sqrt{2x + 1} \)?
5. What are the advantages of using the definition of a derivative over shortcut rules?

**Tip**: Always try to simplify the expressions before plugging in the limit for \( h \to 0 \) when using the definition of a derivative.

Given f(x) = √x − 6, find f′(7) using the definition of a derivative.

Discover how to find the derivative of the function f(x) = √x - 6 at x = 7 using the definition of a derivative. This step-by-step guide covers the limit definition, simplification techniques, and how to apply them to square roots. Suitable for students in Grades 11-12 or introductory college calculus.

Math Problem Statement

Given
f ( x )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation