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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 3.9.19
Chapter 3, Problem 3.9.19

Find the derivative of the following functions.
y = In |sin x|

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Step 1: Recognize that the function y = ln|sin(x)| is a composition of functions, specifically the natural logarithm function and the absolute value of the sine function. To find the derivative, we will use the chain rule.
Step 2: Apply the chain rule. The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, f(u) = ln(u) and g(x) = |sin(x)|.
Step 3: Differentiate the outer function f(u) = ln(u) with respect to u. The derivative of ln(u) is 1/u. So, f'(u) = 1/u.
Step 4: Differentiate the inner function g(x) = |sin(x)| with respect to x. The derivative of |sin(x)| is (sin(x)/|sin(x)|) * cos(x), using the derivative of the absolute value function and the derivative of sin(x).
Step 5: Combine the results from steps 3 and 4 using the chain rule. The derivative of y = ln|sin(x)| is y' = (1/|sin(x)|) * (sin(x)/|sin(x)|) * cos(x). Simplify the expression to get the final 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 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 used in various applications, including optimization and motion analysis.
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Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is essential when dealing with complex functions, such as logarithmic or trigonometric functions combined with others.
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Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of other functions, especially when they involve exponentials or logarithms. By taking the natural logarithm of both sides of an equation, one can simplify the differentiation process, making it easier to apply the product and chain rules. This method is particularly useful for functions like y = ln|sin x|, where the argument is a function itself.
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Related Practice
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