Textbook Question
Find an equation of the line tangent to the given curve at a.
y = (x + 5) / (x - 1); a = 3
3. Techniques of Differentiation
Product and Quotient Rules
5:10 minutesProblem 3.4.37
Textbook Question
Derivatives Find and simplify the derivative of the following functions.
g(x) = e^x / x²-1
Verified step by step guidance1
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.
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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.
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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).
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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).
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