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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.32
Chapter 3, Problem 3.5.32

23–51. Calculating derivatives Find the derivative of the following functions.
y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants

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Step 1: Identify the function for which you need to find the derivative. The function given is \( y = \frac{a \sin x + b \cos x}{a \sin x - b \cos x} \).
Step 2: Recognize that this is a quotient of two functions, so you will need to use the Quotient Rule for differentiation. The Quotient Rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then the derivative \( y' \) is given by \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
Step 3: Identify \( u(x) = a \sin x + b \cos x \) and \( v(x) = a \sin x - b \cos x \). Compute the derivatives \( u'(x) \) and \( v'(x) \). Use the derivatives of sine and cosine: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).
Step 4: Calculate \( u'(x) = a \cos x - b \sin x \) and \( v'(x) = a \cos x + b \sin x \).
Step 5: Substitute \( u(x) \), \( v(x) \), \( u'(x) \), and \( v'(x) \) into the Quotient Rule formula to find the derivative of \( y \). Simplify the expression if possible.

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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 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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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In calculus, these functions are essential for modeling oscillatory behavior and are frequently encountered in problems involving derivatives. Understanding their properties, such as their derivatives, is crucial for solving calculus problems involving these functions.
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Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. It states that if you have a function y = u/v, where u and v are both differentiable functions, the derivative is given by (v * u' - u * v') / v^2. This rule is particularly useful when differentiating functions like the one in the question, where the function is expressed as a fraction.
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Related Practice
Textbook Question

Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. <MATCH A-D IMAGE>

Textbook Question

51–56. Second derivatives Find d²y/dx².

x⁴+y⁴ = 64

Textbook Question

The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?

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.

g (x) = x^ In x; a = e

Textbook Question

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim x🠂2) 1/x+1 - 1/3 / x-2

Textbook Question

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = x⁸cos³ x / √x-1

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