Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals4h 44m
- 9. Graphical Applications of Integrals2h 27m
- 10. Physics Applications of Integrals 2h 22m
3. Techniques of Differentiation
Product and Quotient Rules
Problem 3.4.37
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
g(x) = e^x / x²-1

1
Step 1: Identify the function to differentiate. The function given is \( g(x) = \frac{e^x}{x^2 - 1} \). This is a quotient of two functions, so we will use the Quotient Rule.
Step 2: Recall the Quotient Rule. The Quotient Rule states that if you have a function \( h(x) = \frac{f(x)}{g(x)} \), then its derivative \( h'(x) \) is given by \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).
Step 3: Identify \( f(x) \) and \( g(x) \) in the function \( g(x) = \frac{e^x}{x^2 - 1} \). Here, \( f(x) = e^x \) and \( g(x) = x^2 - 1 \).
Step 4: Differentiate \( f(x) \) and \( g(x) \). The derivative of \( f(x) = e^x \) is \( f'(x) = e^x \). The derivative of \( g(x) = x^2 - 1 \) is \( g'(x) = 2x \).
Step 5: Apply the Quotient Rule. Substitute \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) into the Quotient Rule formula: \( g'(x) = \frac{e^x(x^2 - 1) - e^x(2x)}{(x^2 - 1)^2} \). Simplify the expression to find the derivative.

This video solution was recommended by our tutors as helpful for the problem above
Video duration:
5mPlay a video:
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
Recommended video:
Derivatives
Quotient Rule
The quotient rule is a specific technique used to differentiate functions that are expressed as the ratio of two other functions. If you have a function g(x) = u(x)/v(x), the derivative g'(x) is given by (u'v - uv')/v², where u' and v' are the derivatives of u and v, respectively. This rule is essential for simplifying the differentiation of functions like g(x) = e^x / (x² - 1).
Recommended video:
The Quotient Rule
Exponential Functions
Exponential functions are functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. The derivative of an exponential function, particularly e^x, is unique because it is equal to itself, making it a crucial function in calculus. Understanding how to differentiate exponential functions is vital when working with more complex expressions that involve them, such as in the given function g(x).
Recommended video:
Exponential Functions
Related Videos
Related Practice