Textbook Question
15–48. Derivatives Find the derivative of the following functions.
f(x) = 2^x/2^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 3.5.39
23–51. Calculating derivatives Find the derivative of the following functions.
y = sin x / 1 + cos x
Verified step by step guidance1
Step 1: Recognize that the function \( y = \frac{\sin x}{1 + \cos x} \) is a quotient of two functions, \( u(x) = \sin x \) and \( v(x) = 1 + \cos x \). To find the derivative, we will use the quotient rule.
Step 2: Recall the quotient rule for derivatives, which states that if \( y = \frac{u(x)}{v(x)} \), then \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
Step 3: Differentiate the numerator \( u(x) = \sin x \). The derivative \( u'(x) = \cos x \).
Step 4: Differentiate the denominator \( v(x) = 1 + \cos x \). The derivative \( v'(x) = -\sin x \).
Step 5: Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the quotient rule formula to find \( y' \).
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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 allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Derivatives
Quotient Rule
The Quotient Rule is a formula used to find the derivative of a function that is the ratio of two other functions. If you have a function y = u/v, where u and v are both differentiable functions, the derivative is given by (v * du/dx - u * dv/dx) / v^2. This rule is essential for differentiating functions like the one in the question.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. Their derivatives are also crucial, as they follow specific rules: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is key to solving problems involving trigonometric functions.
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Related Practice
Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
y = xe^y
Textbook Question
If f is continuous at a, must f be differentiable at a?
Textbook Question
Find the derivative of the following functions.
y = sin x + cos x
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Textbook Question
A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft³/min) if the water level is dropping at 6 min/in?
Textbook Question
49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
f (x) = (4 sin x+2)^cos x; a = π
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