Given the function f, of, x, equals, start fraction, 5, divided by, 3, x, end fraction, commaf(x)= 
3x
5
​
 , determine the slope of the tangent line of ff at x, equals, minus, 5x=−5 using the limit shown below. You do not have to simplify your answer.

To determine the slope of the tangent line of the function \( f(x) = \frac{5}{3x} \) at \( x = -5 \), we can use the definition of the derivative. The slope of the tangent line at a point \( x = a \) is given by the limit:

\[
\text{slope} = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\]

For this problem, the function is \( f(x) = \frac{5}{3x} \), and the value of \( a \) is \( -5 \). Substituting these into the limit definition:

\[
\text{slope} = \lim_{h \to 0} \frac{f(-5 + h) - f(-5)}{h}
\]

### Step 1: Calculate \( f(-5 + h) \)
Substitute \( x = -5 + h \) into the function:

\[
f(-5 + h) = \frac{5}{3(-5 + h)} = \frac{5}{-15 + 3h}
\]

### Step 2: Calculate \( f(-5) \)
Substitute \( x = -5 \) into the function:

\[
f(-5) = \frac{5}{3(-5)} = \frac{5}{-15} = -\frac{1}{3}
\]

### Step 3: Set up the limit expression
Now substitute the expressions for \( f(-5 + h) \) and \( f(-5) \) into the limit:

\[
\text{slope} = \lim_{h \to 0} \frac{\frac{5}{-15 + 3h} - \left(-\frac{1}{3}\right)}{h}
\]

This is the expression you need to compute for the slope of the tangent line at \( x = -5 \). You do not need to simplify it further as requested.

Would you like me to simplify this further or clarify any part of the process? 

Here are some related questions you might find useful:
1. What is the general formula for the derivative using limits?
2. How would you compute the slope of the tangent line for other functions?
3. How do you interpret the slope of the tangent line geometrically?
4. What happens to the slope as \( h \to 0 \) for different functions?
5. How would you apply this method to non-fractional functions?

**Tip**: When working with limits involving fractions, be cautious of potential algebraic simplifications to make the process smoother.

Given the function f(x) = \frac{5}{3x}, determine the slope of the tangent line of f at x = -5 using the limit shown below. You do not have to simplify your answer.

Learn how to determine the slope of the tangent line of the function f(x) = 5/(3x) at x = -5 using the limit definition of the derivative. This step-by-step guide explains key calculus concepts like limits and derivatives. 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