Textbook Question
9–61. Evaluate and simplify y'.
y = e^tan x (tan x−1)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.R.84C
Use the given graphs of f and g to find each derivative. <IMAGE>
c. d/dx ((f(x) / g(x)) |x=3
Verified step by step guidance1
To find the derivative of the function \( \frac{f(x)}{g(x)} \) at \( x = 3 \), we will use 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 \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).
First, identify \( f(x) \) and \( g(x) \) from the graphs at \( x = 3 \). Determine the values of \( f(3) \) and \( g(3) \) from the graph.
Next, find the derivatives \( f'(x) \) and \( g'(x) \) at \( x = 3 \). This involves determining the slopes of the tangent lines to the graphs of \( f \) and \( g \) at \( x = 3 \).
Substitute the values \( f(3) \), \( g(3) \), \( f'(3) \), and \( g'(3) \) into the Quotient Rule formula: \( h'(3) = \frac{f'(3)g(3) - f(3)g'(3)}{(g(3))^2} \).
Simplify the expression obtained from the Quotient Rule to find the derivative of \( \frac{f(x)}{g(x)} \) at \( x = 3 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are denoted as f'(x) or df/dx.
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Derivatives
Quotient Rule
The Quotient Rule is a formula used to find the derivative of a function that is the quotient of two other functions. If h(x) = f(x) / g(x), the derivative h'(x) is given by h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. This rule is essential when differentiating functions that are divided by one another.
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06:43The Quotient Rule
Evaluating Derivatives at a Point
Evaluating a derivative at a specific point involves substituting the x-value into the derivative function. For example, to find d/dx (f(x) / g(x)) at x=3, you first apply the Quotient Rule to find the derivative and then substitute x=3 into the resulting expression. This process provides the instantaneous rate of change of the function at that particular point.
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Related Practice
Textbook Question
Use the given graphs of f and g to find each derivative. <IMAGE>
b. d/dx (f(x)g(x)) |x=1
Textbook Question
66–71. Higher-order derivatives Find and simplify y''.
x + sin y = y
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Textbook Question
Evaluate and simplify y'.
y = ln |sec 3x|
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Textbook Question
{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.
b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.
Textbook Question
Use differentiation to verify each equation.
d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).