Using the definition of the derivative, find f(x). Then find f'( – 6), f'(0), and f'(5) when the derivative exists. 72 X f(x) f'(x)= (Type an expression using x as the variable.) .…..
Using the definition of the derivative, find f(x). Then find f'( – 6), f'(0), and f'(5) when the derivative exists. 72 X f(x) f'(x)= (Type an expression using x as the variable.) .…..
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Finding Derivatives Using the Definition of the Derivative**
**Introduction:**
In this tutorial, we will learn how to calculate the derivative of a function using the definition of the derivative. We will apply this method to the given function \( f(x) = \frac{72}{x} \). Additionally, we will find the derivative at specific points: \( f'(-6) \), \( f'(0) \), and \( f'(5) \), provided the derivative exists at those points.
**Problem Statement:**
Given the function:
\[ f(x) = \frac{72}{x} \]
1. Use the definition of the derivative to find \( f'(x) \).
2. Calculate \( f'(-6) \), \( f'(0) \), and \( f'(5) \), if the derivative exists at these points.
**Step-by-Step Solution:**
1. **Find \( f'(x) \) Using the Definition of the Derivative:**
The definition of the derivative is given by:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]
For the function \( f(x) = \frac{72}{x} \), substitute \( f(x) \) and \( f(x+h) \):
\[ f'(x) = \lim_{{h \to 0}} \frac{\frac{72}{x+h} - \frac{72}{x}}{h} \]
Simplify the expression inside the limit by finding a common denominator for the fractions:
\[ f'(x) = \lim_{{h \to 0}} \frac{72(x) - 72(x+h)}{h(x(x+h))} \]
\[ f'(x) = \lim_{{h \to 0}} \frac{72x - 72x - 72h}{h(x^2 + xh)} \]
\[ f'(x) = \lim_{{h \to 0}} \frac{-72h}{h(x^2 + xh)} \]
Cancel \( h \) in the numerator and denominator:
\[ f'(x) = \lim_{{h \to 0}} \frac{-72}{x^2 + xh} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdac7294-121d-461c-b626-bf307098b471%2F9b01e9d7-3b47-4d2c-aff7-8510554987a4%2Fthlk3qt_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Finding Derivatives Using the Definition of the Derivative**
**Introduction:**
In this tutorial, we will learn how to calculate the derivative of a function using the definition of the derivative. We will apply this method to the given function \( f(x) = \frac{72}{x} \). Additionally, we will find the derivative at specific points: \( f'(-6) \), \( f'(0) \), and \( f'(5) \), provided the derivative exists at those points.
**Problem Statement:**
Given the function:
\[ f(x) = \frac{72}{x} \]
1. Use the definition of the derivative to find \( f'(x) \).
2. Calculate \( f'(-6) \), \( f'(0) \), and \( f'(5) \), if the derivative exists at these points.
**Step-by-Step Solution:**
1. **Find \( f'(x) \) Using the Definition of the Derivative:**
The definition of the derivative is given by:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]
For the function \( f(x) = \frac{72}{x} \), substitute \( f(x) \) and \( f(x+h) \):
\[ f'(x) = \lim_{{h \to 0}} \frac{\frac{72}{x+h} - \frac{72}{x}}{h} \]
Simplify the expression inside the limit by finding a common denominator for the fractions:
\[ f'(x) = \lim_{{h \to 0}} \frac{72(x) - 72(x+h)}{h(x(x+h))} \]
\[ f'(x) = \lim_{{h \to 0}} \frac{72x - 72x - 72h}{h(x^2 + xh)} \]
\[ f'(x) = \lim_{{h \to 0}} \frac{-72h}{h(x^2 + xh)} \]
Cancel \( h \) in the numerator and denominator:
\[ f'(x) = \lim_{{h \to 0}} \frac{-72}{x^2 + xh} \
![For the given function, find (a) the equation of the secant line through the points where \( x \) has the given values and (b) the equation of the tangent line when \( x \) has the first value.
\[ y = f(x) = x^2 + x; \quad x = 3, \quad x = 5 \]
---
**a.** The equation of the secant line is \( y = \_\_\_ \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdac7294-121d-461c-b626-bf307098b471%2F9b01e9d7-3b47-4d2c-aff7-8510554987a4%2Fvh4lct9_processed.png&w=3840&q=75)
Transcribed Image Text:For the given function, find (a) the equation of the secant line through the points where \( x \) has the given values and (b) the equation of the tangent line when \( x \) has the first value.
\[ y = f(x) = x^2 + x; \quad x = 3, \quad x = 5 \]
---
**a.** The equation of the secant line is \( y = \_\_\_ \)
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