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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 3.5.47
Chapter 3, Problem 3.5.47

Find the derivative of the following functions.
y = cot x / (1 + csc x)

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1
Step 1: Identify the functions involved. The given function is \( y = \frac{\cot x}{1 + \csc x} \). This is a quotient of two functions: the numerator \( u = \cot x \) and the denominator \( v = 1 + \csc x \).
Step 2: Apply the Quotient Rule for derivatives. The Quotient Rule states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \).
Step 3: Find the derivative of the numerator \( u = \cot x \). The derivative \( u' = -\csc^2 x \).
Step 4: Find the derivative of the denominator \( v = 1 + \csc x \). The derivative \( v' = -\csc x \cot x \).
Step 5: Substitute \( u' \), \( v \), \( u \), and \( v' \) into the Quotient Rule formula: \( y' = \frac{(-\csc^2 x)(1 + \csc x) - (\cot x)(-\csc x \cot x)}{(1 + \csc x)^2} \). 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.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx and can be calculated using various rules, such as the product rule, quotient rule, and chain rule.
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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 of x, the derivative is given by y' = (v * u' - u * v') / v^2. This rule is essential for differentiating functions like the one in the question, where cot x is divided by (1 + csc x).
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The Quotient Rule

Trigonometric Functions and Their Derivatives

Trigonometric functions such as cotangent (cot) and cosecant (csc) have specific derivatives that are crucial for solving problems involving these functions. The derivative of cot x is -csc^2 x, and the derivative of csc x is -csc x * cot x. Understanding these derivatives allows for the effective application of the quotient rule and the overall differentiation process in the given function.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Find the following higher-order derivatives.

dn/dxn (2x)

Textbook Question

Find and simplify the derivative of the following functions.

f(x) = ex(x3 − 3x2 + 6x − 6)

Textbook Question

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x) = 1/2x+8; (10,4)

Textbook Question

73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>


Exercise 46

Textbook Question

Evaluate the derivative of the following functions.

f(x) = sin(tan-1 (ln x))

Textbook Question

Consider the line f(x)=mx+b, where m and b are constants. Show that f′(x)=m for all x. Interpret this result.